**Authors**
Gerald Jay Sussman
Harold Abelson
Julie Sussman

**License**
CC-BY-SA-4.0

Structure and Interpretation of Computer Programs second edition Unofﬁcial Texinfo Format 2.andresraba5.6 Harold Abelson and Gerald Jay Sussman with Julie Sussman foreword by Alan J. Perlis ©1996 by e Massachuses Institute of Technology Structure and Interpretation of Computer Programs, second edition Harold Abelson and Gerald Jay Sussman with Julie Sussman, foreword by Alan J. Perlis is work is licensed under a Creative Commons Aribution-ShareAlike 4.0 International License ( .). Based on a work at mitpress.mit.edu. e Press Cambridge, Massachuses London, England McGraw-Hill Book Company New York, St. Louis, San Francisco, Montreal, Toronto Unoﬃcial Texinfo Format 2.andresraba5.6 (February 2, 2016), based on 2.neilvandyke4 (January 10, 2007). Contents Unoﬃcial Texinfo Format ix Dedication xii Foreword xiii Preface to the Second Edition xix Preface to the First Edition xxi Anowledgments xxv 1 Building Abstractions with Procedures 1 1.1 e Elements of Programming . . . . . . . . . . . . . . 6 1.1.1 Expressions . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Naming and the Environment . . . . . . . . . . 10 1.1.3 Evaluating Combinations . . . . . . . . . . . . 12 1.1.4 Compound Procedures . . . . . . . . . . . . . . 15 1.1.5 e Substitution Model for Procedure Application 18 1.1.6 Conditional Expressions and Predicates . . . . 22 1.1.7 Example: Square Roots by Newton’s Method . . 28 iii 1.1.8 Procedures as Black-Box Abstractions . . . . . 33 1.2 Procedures and the Processes ey Generate . . . . . . 40 1.2.1 Linear Recursion and Iteration . . . . . . . . . 41 1.2.2 Tree Recursion . . . . . . . . . . . . . . . . . . 47 1.2.3 Orders of Growth . . . . . . . . . . . . . . . . . 54 1.2.4 Exponentiation . . . . . . . . . . . . . . . . . . 57 1.2.5 Greatest Common Divisors . . . . . . . . . . . 62 1.2.6 Example: Testing for Primality . . . . . . . . . 65 1.3 Formulating Abstractions with Higher-Order Procedures . . . . . . . . . . . . . . 74 1.3.1 Procedures as Arguments . . . . . . . . . . . . 76 1.3.2 Constructing Procedures Using lambda . . . . . 83 1.3.3 Procedures as General Methods . . . . . . . . . 89 1.3.4 Procedures as Returned Values . . . . . . . . . 97 2 Building Abstractions with Data 107 2.1 Introduction to Data Abstraction . . . . . . . . . . . . . 112 2.1.1 Example: Arithmetic Operations for Rational Numbers . . . . . . . . . . . . . . . 113 2.1.2 Abstraction Barriers . . . . . . . . . . . . . . . 118 2.1.3 What Is Meant by Data? . . . . . . . . . . . . . 122 2.1.4 Extended Exercise: Interval Arithmetic . . . . . 126 2.2 Hierarchical Data and the Closure Property . . . . . . . 132 2.2.1 Representing Sequences . . . . . . . . . . . . . 134 2.2.2 Hierarchical Structures . . . . . . . . . . . . . . 147 2.2.3 Sequences as Conventional Interfaces . . . . . 154 2.2.4 Example: A Picture Language . . . . . . . . . . 172 2.3 Symbolic Data . . . . . . . . . . . . . . . . . . . . . . . 192 2.3.1 otation . . . . . . . . . . . . . . . . . . . . . 192 iv 2.3.2 Example: Symbolic Diﬀerentiation . . . . . . . 197 2.3.3 Example: Representing Sets . . . . . . . . . . . 205 2.3.4 Example: Huﬀman Encoding Trees . . . . . . . 218 2.4 Multiple Representations for Abstract Data . . . . . . . 229 2.4.1 Representations for Complex Numbers . . . . . 232 2.4.2 Tagged data . . . . . . . . . . . . . . . . . . . . 237 2.4.3 Data-Directed Programming and Additivity . . 242 2.5 Systems with Generic Operations . . . . . . . . . . . . 254 2.5.1 Generic Arithmetic Operations . . . . . . . . . 255 2.5.2 Combining Data of Diﬀerent Types . . . . . . . 262 2.5.3 Example: Symbolic Algebra . . . . . . . . . . . 274 3 Modularity, Objects, and State 294 3.1 Assignment and Local State . . . . . . . . . . . . . . . 296 3.1.1 Local State Variables . . . . . . . . . . . . . . . 297 3.1.2 e Beneﬁts of Introducing Assignment . . . . 305 3.1.3 e Costs of Introducing Assignment . . . . . . 311 3.2 e Environment Model of Evaluation . . . . . . . . . . 320 3.2.1 e Rules for Evaluation . . . . . . . . . . . . . 322 3.2.2 Applying Simple Procedures . . . . . . . . . . . 327 3.2.3 Frames as the Repository of Local State . . . . 330 3.2.4 Internal Deﬁnitions . . . . . . . . . . . . . . . . 337 3.3 Modeling with Mutable Data . . . . . . . . . . . . . . . 341 3.3.1 Mutable List Structure . . . . . . . . . . . . . . 342 3.3.2 Representing eues . . . . . . . . . . . . . . . 353 3.3.3 Representing Tables . . . . . . . . . . . . . . . 360 3.3.4 A Simulator for Digital Circuits . . . . . . . . . 369 3.3.5 Propagation of Constraints . . . . . . . . . . . 386 3.4 Concurrency: Time Is of the Essence . . . . . . . . . . . 401 v 3.4.1 e Nature of Time in Concurrent Systems . . 403 3.4.2 Mechanisms for Controlling Concurrency . . . 410 3.5 Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 3.5.1 Streams Are Delayed Lists . . . . . . . . . . . . 430 3.5.2 Inﬁnite Streams . . . . . . . . . . . . . . . . . . 441 3.5.3 Exploiting the Stream Paradigm . . . . . . . . . 453 3.5.4 Streams and Delayed Evaluation . . . . . . . . 470 3.5.5 Modularity of Functional Programs and Modularity of Objects . . . . . . . . . . . . 479 4 Metalinguistic Abstraction 487 4.1 e Metacircular Evaluator . . . . . . . . . . . . . . . . 492 4.1.1 e Core of the Evaluator . . . . . . . . . . . . 495 4.1.2 Representing Expressions . . . . . . . . . . . . 501 4.1.3 Evaluator Data Structures . . . . . . . . . . . . 512 4.1.4 Running the Evaluator as a Program . . . . . . 518 4.1.5 Data as Programs . . . . . . . . . . . . . . . . . 522 4.1.6 Internal Deﬁnitions . . . . . . . . . . . . . . . . 526 4.1.7 Separating Syntactic Analysis from Execution . 534 4.2 Variations on a Scheme — Lazy Evaluation . . . . . . . 541 4.2.1 Normal Order and Applicative Order . . . . . . 542 4.2.2 An Interpreter with Lazy Evaluation . . . . . . 544 4.2.3 Streams as Lazy Lists . . . . . . . . . . . . . . . 555 4.3 Variations on a Scheme — Nondeterministic Computing 559 4.3.1 Amb and Search . . . . . . . . . . . . . . . . . 561 4.3.2 Examples of Nondeterministic Programs . . . . 567 4.3.3 Implementing the amb Evaluator . . . . . . . . 578 4.4 Logic Programming . . . . . . . . . . . . . . . . . . . . 594 4.4.1 Deductive Information Retrieval . . . . . . . . 599 vi 4.4.2 How the ery System Works . . . . . . . . . 615 4.4.3 Is Logic Programming Mathematical Logic? . . 627 4.4.4 Implementing the ery System . . . . . . . . 635 4.4.4.1 e Driver Loop and Instantiation . . 636 4.4.4.2 e Evaluator . . . . . . . . . . . . . 638 4.4.4.3 Finding Assertions by Paern Matching . . . . . . . . . 642 4.4.4.4 Rules and Uniﬁcation . . . . . . . . . 645 4.4.4.5 Maintaining the Data Base . . . . . . 651 4.4.4.6 Stream Operations . . . . . . . . . . 654 4.4.4.7 ery Syntax Procedures . . . . . . . 656 4.4.4.8 Frames and Bindings . . . . . . . . . 659 5 Computing with Register Maines 666 5.1 Designing Register Machines . . . . . . . . . . . . . . . 668 5.1.1 A Language for Describing Register Machines . 672 5.1.2 Abstraction in Machine Design . . . . . . . . . 678 5.1.3 Subroutines . . . . . . . . . . . . . . . . . . . . 681 5.1.4 Using a Stack to Implement Recursion . . . . . 686 5.1.5 Instruction Summary . . . . . . . . . . . . . . . 695 5.2 A Register-Machine Simulator . . . . . . . . . . . . . . 696 5.2.1 e Machine Model . . . . . . . . . . . . . . . . 698 5.2.2 e Assembler . . . . . . . . . . . . . . . . . . 704 5.2.3 Generating Execution Procedures for Instructions . . . . . . . . . . . . . . . . . . 708 5.2.4 Monitoring Machine Performance . . . . . . . 718 5.3 Storage Allocation and Garbage Collection . . . . . . . 723 5.3.1 Memory as Vectors . . . . . . . . . . . . . . . . 724 5.3.2 Maintaining the Illusion of Inﬁnite Memory . . 731 vii 5.4 e Explicit-Control Evaluator . . . . . . . . . . . . . . 741 5.4.1 e Core of the Explicit-Control Evaluator . . . 743 5.4.2 Sequence Evaluation and Tail Recursion . . . . 751 5.4.3 Conditionals, Assignments, and Deﬁnitions . . 756 5.4.4 Running the Evaluator . . . . . . . . . . . . . . 759 5.5 Compilation . . . . . . . . . . . . . . . . . . . . . . . . 767 5.5.1 Structure of the Compiler . . . . . . . . . . . . 772 5.5.2 Compiling Expressions . . . . . . . . . . . . . . 779 5.5.3 Compiling Combinations . . . . . . . . . . . . 788 5.5.4 Combining Instruction Sequences . . . . . . . . 797 5.5.5 An Example of Compiled Code . . . . . . . . . 802 5.5.6 Lexical Addressing . . . . . . . . . . . . . . . . 817 5.5.7 Interfacing Compiled Code to the Evaluator . . 823 References 834 List of Exercises 844 List of Figures 846 Index 848 Colophon 855 viii Unoﬀicial Texinfo Format is is the second edition book, from Unoﬃcial Texinfo Format. You are probably reading it in an Info hypertext browser, such as the Info mode of Emacs. You might alternatively be reading it TEX-formaed on your screen or printer, though that would be silly. And, if printed, expensive. e freely-distributed oﬃcial -and- format was ﬁrst con- verted personally to Unoﬃcial Texinfo Format () version 1 by Lytha Ayth during a long Emacs lovefest weekend in April, 2001. e is easier to search than the format. It is also much more accessible to people running on modest computers, such as do- nated ’386-based PCs. A 386 can, in theory, run Linux, Emacs, and a Scheme interpreter simultaneously, but most 386s probably can’t also run both Netscape and the necessary X Window System without prema- turely introducing budding young underfunded hackers to the concept of thrashing. can also ﬁt uncompressed on a 1.44 ﬂoppy diskee, which may come in handy for installing on PCs that do not have Internet or access. e Texinfo conversion has been a straight transliteration, to the extent possible. Like the TEX-to- conversion, this was not without some introduction of breakage. In the case of Unoﬃcial Texinfo Format, ix ﬁgures have suﬀered an amateurish resurrection of the lost art of . Also, it’s quite possible that some errors of ambiguity were introduced during the conversion of some of the copious superscripts (‘ˆ’) and sub- scripts (‘_’). Divining which has been le as an exercise to the reader. But at least we don’t put our brave astronauts at risk by encoding the greater-than-or-equal symbol as <u>></u>. If you modify sicp.texi to correct errors or improve the art, then update the @set utfversion 2.andresraba5.6 line to reﬂect your delta. For example, if you started with Lytha’s version 1, and your name is Bob, then you could name your successive versions 1.bob1, 1.bob2, . . . 1.bobn. Also update utfversiondate. If you want to distribute your version on the Web, then embedding the string “sicp.texi” somewhere in the ﬁle or Web page will make it easier for people to ﬁnd with Web search engines. It is believed that the Unoﬃcial Texinfo Format is in keeping with the spirit of the graciously freely-distributed version. But you never know when someone’s armada of lawyers might need something to do, and get their shorts all in a knot over some benign lile thing, so think twice before you use your full name or distribute Info, , PostScript, or formats that might embed your account or machine name. Peath, Lytha Ayth Addendum: See also the video lectures by Abelson and Sussman: at or . Second Addendum: Above is the original introduction to the from 2001. Ten years later, has been transformed: mathematical symbols and formulas are properly typeset, and ﬁgures drawn in vector graph- ics. e original text formulas and art ﬁgures are still there in x the Texinfo source, but will display only when compiled to Info output. At the dawn of e-book readers and tablets, reading a on screen is oﬃcially not silly anymore. Enjoy! A.R, May, 2011 xi Dedication T , in respect and admiration, to the spirit that lives in the computer. “I think that it’s extraordinarily important that we in com- puter science keep fun in computing. When it started out, it was an awful lot of fun. Of course, the paying customers got shaed every now and then, and aer a while we began to take their complaints seriously. We began to feel as if we really were responsible for the successful, error-free perfect use of these machines. I don’t think we are. I think we’re responsible for stretching them, seing them oﬀ in new di- rections, and keeping fun in the house. I hope the ﬁeld of computer science never loses its sense of fun. Above all, I hope we don’t become missionaries. Don’t feel as if you’re Bible salesmen. e world has too many of those already. What you know about computing other people will learn. Don’t feel as if the key to successful computing is only in your hands. What’s in your hands, I think and hope, is in- telligence: the ability to see the machine as more than when you were ﬁrst led up to it, that you can make it more.” —Alan J. Perlis (April 1, 1922 – February 7, 1990) xii Foreword E , , , psychologists, and parents pro- gram. Armies, students, and some societies are programmed. An assault on large problems employs a succession of programs, most of which spring into existence en route. ese programs are rife with is- sues that appear to be particular to the problem at hand. To appreciate programming as an intellectual activity in its own right you must turn to computer programming; you must read and write computer programs— many of them. It doesn’t maer much what the programs are about or what applications they serve. What does maer is how well they per- form and how smoothly they ﬁt with other programs in the creation of still greater programs. e programmer must seek both perfection of part and adequacy of collection. In this book the use of “program” is focused on the creation, execution, and study of programs wrien in a dialect of Lisp for execution on a digital computer. Using Lisp we re- strict or limit not what we may program, but only the notation for our program descriptions. Our traﬃc with the subject maer of this book involves us with three foci of phenomena: the human mind, collections of computer pro- grams, and the computer. Every computer program is a model, hatched in the mind, of a real or mental process. ese processes, arising from xiii human experience and thought, are huge in number, intricate in de- tail, and at any time only partially understood. ey are modeled to our permanent satisfaction rarely by our computer programs. us even though our programs are carefully handcraed discrete collections of symbols, mosaics of interlocking functions, they continually evolve: we change them as our perception of the model deepens, enlarges, gen- eralizes until the model ultimately aains a metastable place within still another model with which we struggle. e source of the exhilara- tion associated with computer programming is the continual unfolding within the mind and on the computer of mechanisms expressed as pro- grams and the explosion of perception they generate. If art interprets our dreams, the computer executes them in the guise of programs! For all its power, the computer is a harsh taskmaster. Its programs must be correct, and what we wish to say must be said accurately in ev- ery detail. As in every other symbolic activity, we become convinced of program truth through argument. Lisp itself can be assigned a seman- tics (another model, by the way), and if a program’s function can be speciﬁed, say, in the predicate calculus, the proof methods of logic can be used to make an acceptable correctness argument. Unfortunately, as programs get large and complicated, as they almost always do, the ade- quacy, consistency, and correctness of the speciﬁcations themselves be- come open to doubt, so that complete formal arguments of correctness seldom accompany large programs. Since large programs grow from small ones, it is crucial that we develop an arsenal of standard program structures of whose correctness we have become sure—we call them idioms—and learn to combine them into larger structures using orga- nizational techniques of proven value. ese techniques are treated at length in this book, and understanding them is essential to participation in the Promethean enterprise called programming. More than anything xiv else, the uncovering and mastery of powerful organizational techniques accelerates our ability to create large, signiﬁcant programs. Conversely, since writing large programs is very taxing, we are stimulated to invent new methods of reducing the mass of function and detail to be ﬁed into large programs. Unlike programs, computers must obey the laws of physics. If they wish to perform rapidly—a few nanoseconds per state change—they must transmit electrons only small distances (at most 1 12 feet). e heat generated by the huge number of devices so concentrated in space has to be removed. An exquisite engineering art has been developed balancing between multiplicity of function and density of devices. In any event, hardware always operates at a level more primitive than that at which we care to program. e processes that transform our Lisp programs to “machine” programs are themselves abstract models which we pro- gram. eir study and creation give a great deal of insight into the or- ganizational programs associated with programming arbitrary models. Of course the computer itself can be so modeled. ink of it: the behav- ior of the smallest physical switching element is modeled by quantum mechanics described by diﬀerential equations whose detailed behavior is captured by numerical approximations represented in computer pro- grams executing on computers composed of . . .! It is not merely a maer of tactical convenience to separately iden- tify the three foci. Even though, as they say, it’s all in the head, this logical separation induces an acceleration of symbolic traﬃc between these foci whose richness, vitality, and potential is exceeded in human experience only by the evolution of life itself. At best, relationships be- tween the foci are metastable. e computers are never large enough or fast enough. Each breakthrough in hardware technology leads to more massive programming enterprises, new organizational principles, and xv an enrichment of abstract models. Every reader should ask himself pe- riodically “Toward what end, toward what end?”—but do not ask it too oen lest you pass up the fun of programming for the constipation of biersweet philosophy. Among the programs we write, some (but never enough) perform a precise mathematical function such as sorting or ﬁnding the maximum of a sequence of numbers, determining primality, or ﬁnding the square root. We call such programs algorithms, and a great deal is known of their optimal behavior, particularly with respect to the two important parameters of execution time and data storage requirements. A pro- grammer should acquire good algorithms and idioms. Even though some programs resist precise speciﬁcations, it is the responsibility of the pro- grammer to estimate, and always to aempt to improve, their perfor- mance. Lisp is a survivor, having been in use for about a quarter of a cen- tury. Among the active programming languages only Fortran has had a longer life. Both languages have supported the programming needs of important areas of application, Fortran for scientiﬁc and engineering computation and Lisp for artiﬁcial intelligence. ese two areas con- tinue to be important, and their programmers are so devoted to these two languages that Lisp and Fortran may well continue in active use for at least another quarter-century. Lisp changes. e Scheme dialect used in this text has evolved from the original Lisp and diﬀers from the laer in several important ways, including static scoping for variable binding and permiing functions to yield functions as values. In its semantic structure Scheme is as closely akin to Algol 60 as to early Lisps. Algol 60, never to be an active language again, lives on in the genes of Scheme and Pascal. It would be diﬃcult to ﬁnd two languages that are the communicating coin of two more dif- xvi ferent cultures than those gathered around these two languages. Pas- cal is for building pyramids—imposing, breathtaking, static structures built by armies pushing heavy blocks into place. Lisp is for building organisms—imposing, breathtaking, dynamic structures built by squads ﬁing ﬂuctuating myriads of simpler organisms into place. e organiz- ing principles used are the same in both cases, except for one extraordi- narily important diﬀerence: e discretionary exportable functionality entrusted to the individual Lisp programmer is more than an order of magnitude greater than that to be found within Pascal enterprises. Lisp programs inﬂate libraries with functions whose utility transcends the application that produced them. e list, Lisp’s native data structure, is largely responsible for such growth of utility. e simple structure and natural applicability of lists are reﬂected in functions that are amazingly nonidiosyncratic. In Pascal the plethora of declarable data structures in- duces a specialization within functions that inhibits and penalizes ca- sual cooperation. It is beer to have 100 functions operate on one data structure than to have 10 functions operate on 10 data structures. As a result the pyramid must stand unchanged for a millennium; the organ- ism must evolve or perish. To illustrate this diﬀerence, compare the treatment of material and exercises within this book with that in any ﬁrst-course text using Pascal. Do not labor under the illusion that this is a text digestible at only, peculiar to the breed found there. It is precisely what a serious book on programming Lisp must be, no maer who the student is or where it is used. Note that this is a text about programming, unlike most Lisp books, which are used as a preparation for work in artiﬁcial intelligence. Aer all, the critical programming concerns of soware engineering and ar- tiﬁcial intelligence tend to coalesce as the systems under investigation xvii become larger. is explains why there is such growing interest in Lisp outside of artiﬁcial intelligence. As one would expect from its goals, artiﬁcial intelligence research generates many signiﬁcant programming problems. In other program- ming cultures this spate of problems spawns new languages. Indeed, in any very large programming task a useful organizing principle is to con- trol and isolate traﬃc within the task modules via the invention of lan- guage. ese languages tend to become less primitive as one approaches the boundaries of the system where we humans interact most oen. As a result, such systems contain complex language-processing functions replicated many times. Lisp has such a simple syntax and semantics that parsing can be treated as an elementary task. us parsing technology plays almost no role in Lisp programs, and the construction of language processors is rarely an impediment to the rate of growth and change of large Lisp systems. Finally, it is this very simplicity of syntax and se- mantics that is responsible for the burden and freedom borne by all Lisp programmers. No Lisp program of any size beyond a few lines can be wrien without being saturated with discretionary functions. Invent and ﬁt; have ﬁts and reinvent! We toast the Lisp programmer who pens his thoughts within nests of parentheses. Alan J. Perlis New Haven, Connecticut xviii Preface to the Second Edition Is it possible that soware is not like anything else, that it is meant to be discarded: that the whole point is to always see it as a soap bubble? —Alan J. Perlis T has been the basis of ’s entry-level computer science subject since 1980. We had been teaching this ma- terial for four years when the ﬁrst edition was published, and twelve more years have elapsed until the appearance of this second edition. We are pleased that our work has been widely adopted and incorpo- rated into other texts. We have seen our students take the ideas and programs in this book and build them in as the core of new computer systems and languages. In literal realization of an ancient Talmudic pun, our students have become our builders. We are lucky to have such ca- pable students and such accomplished builders. In preparing this edition, we have incorporated hundreds of clariﬁ- cations suggested by our own teaching experience and the comments of colleagues at and elsewhere. We have redesigned most of the ma- jor programming systems in the book, including the generic-arithmetic system, the interpreters, the register-machine simulator, and the com- piler; and we have rewrien all the program examples to ensure that xix any Scheme implementation conforming to the Scheme standard (IEEE 1990) will be able to run the code. is edition emphasizes several new themes. e most important of these is the central role played by diﬀerent approaches to dealing with time in computational models: objects with state, concurrent program- ming, functional programming, lazy evaluation, and nondeterministic programming. We have included new sections on concurrency and non- determinism, and we have tried to integrate this theme throughout the book. e ﬁrst edition of the book closely followed the syllabus of our one-semester subject. With all the new material in the second edi- tion, it will not be possible to cover everything in a single semester, so the instructor will have to pick and choose. In our own teaching, we sometimes skip the section on logic programming (Section 4.4), we have students use the register-machine simulator but we do not cover its im- plementation (Section 5.2), and we give only a cursory overview of the compiler (Section 5.5). Even so, this is still an intense course. Some in- structors may wish to cover only the ﬁrst three or four chapters, leaving the other material for subsequent courses. e World-Wide-Web site hp://mitpress.mit.edu/sicp provides sup- port for users of this book. is includes programs from the book, sam- ple programming assignments, supplementary materials, and download- able implementations of the Scheme dialect of Lisp. xx Preface to the First Edition A computer is like a violin. You can imagine a novice try- ing ﬁrst a phonograph and then a violin. e laer, he says, sounds terrible. at is the argument we have heard from our humanists and most of our computer scientists. Com- puter programs are good, they say, for particular purposes, but they aren’t ﬂexible. Neither is a violin, or a typewriter, until you learn how to use it. —Marvin Minsky, “Why Programming Is a Good Medium for Expressing Poorly-Understood and Sloppily-Formulated Ideas” “T S I C P” is the entry-level subject in computer science at the Massachuses Institute of Technology. It is required of all students at who major in electrical engineering or in computer science, as one-fourth of the “common core curriculum,” which also includes two subjects on circuits and linear systems and a subject on the design of digital systems. We have been involved in the development of this subject since 1978, and we have taught this material in its present form since the fall of 1980 to between 600 and 700 students each year. Most of these students have xxi had lile or no prior formal training in computation, although many have played with computers a bit and a few have had extensive pro- gramming or hardware-design experience. Our design of this introductory computer-science subject reﬂects two major concerns. First, we want to establish the idea that a com- puter language is not just a way of geing a computer to perform oper- ations but rather that it is a novel formal medium for expressing ideas about methodology. us, programs must be wrien for people to read, and only incidentally for machines to execute. Second, we believe that the essential material to be addressed by a subject at this level is not the syntax of particular programming-language constructs, nor clever algorithms for computing particular functions eﬃciently, nor even the mathematical analysis of algorithms and the foundations of computing, but rather the techniques used to control the intellectual complexity of large soware systems. Our goal is that students who complete this subject should have a good feel for the elements of style and the aesthetics of programming. ey should have command of the major techniques for controlling complexity in a large system. ey should be capable of reading a 50- page-long program, if it is wrien in an exemplary style. ey should know what not to read, and what they need not understand at any mo- ment. ey should feel secure about modifying a program, retaining the spirit and style of the original author. ese skills are by no means unique to computer programming. e techniques we teach and draw upon are common to all of engineering design. We control complexity by building abstractions that hide details when appropriate. We control complexity by establishing conventional interfaces that enable us to construct systems by combining standard, well-understood pieces in a “mix and match” way. We control complex- xxii ity by establishing new languages for describing a design, each of which emphasizes particular aspects of the design and deemphasizes others. Underlying our approach to this subject is our conviction that “com- puter science” is not a science and that its signiﬁcance has lile to do with computers. e computer revolution is a revolution in the way we think and in the way we express what we think. e essence of this change is the emergence of what might best be called procedural epis- temology —the study of the structure of knowledge from an imperative point of view, as opposed to the more declarative point of view taken by classical mathematical subjects. Mathematics provides a framework for dealing precisely with notions of “what is.” Computation provides a framework for dealing precisely with notions of “how to.” In teaching our material we use a dialect of the programming lan- guage Lisp. We never formally teach the language, because we don’t have to. We just use it, and students pick it up in a few days. is is one great advantage of Lisp-like languages: ey have very few ways of forming compound expressions, and almost no syntactic structure. All of the formal properties can be covered in an hour, like the rules of chess. Aer a short time we forget about syntactic details of the lan- guage (because there are none) and get on with the real issues—ﬁguring out what we want to compute, how we will decompose problems into manageable parts, and how we will work on the parts. Another advan- tage of Lisp is that it supports (but does not enforce) more of the large- scale strategies for modular decomposition of programs than any other language we know. We can make procedural and data abstractions, we can use higher-order functions to capture common paerns of usage, we can model local state using assignment and data mutation, we can link parts of a program with streams and delayed evaluation, and we can easily implement embedded languages. All of this is embedded in an in- xxiii teractive environment with excellent support for incremental program design, construction, testing, and debugging. We thank all the genera- tions of Lisp wizards, starting with John McCarthy, who have fashioned a ﬁne tool of unprecedented power and elegance. Scheme, the dialect of Lisp that we use, is an aempt to bring to- gether the power and elegance of Lisp and Algol. From Lisp we take the metalinguistic power that derives from the simple syntax, the uniform representation of programs as data objects, and the garbage-collected heap-allocated data. From Algol we take lexical scoping and block struc- ture, which are gis from the pioneers of programming-language de- sign who were on the Algol commiee. We wish to cite John Reynolds and Peter Landin for their insights into the relationship of Church’s λ- calculus to the structure of programming languages. We also recognize our debt to the mathematicians who scouted out this territory decades before computers appeared on the scene. ese pioneers include Alonzo Church, Barkley Rosser, Stephen Kleene, and Haskell Curry. xxiv Acknowledgments W the many people who have helped us develop this book and this curriculum. Our subject is a clear intellectual descendant of “6.231,” a wonderful subject on programming linguistics and the λ-calculus taught at in the late 1960s by Jack Wozencra and Arthur Evans, Jr. We owe a great debt to Robert Fano, who reorganized ’s intro- ductory curriculum in electrical engineering and computer science to emphasize the principles of engineering design. He led us in starting out on this enterprise and wrote the ﬁrst set of subject notes from which this book evolved. Much of the style and aesthetics of programming that we try to teach were developed in conjunction with Guy Lewis Steele Jr., who collaborated with Gerald Jay Sussman in the initial development of the Scheme language. In addition, David Turner, Peter Henderson, Dan Fried- man, David Wise, and Will Clinger have taught us many of the tech- niques of the functional programming community that appear in this book. Joel Moses taught us about structuring large systems. His experi- ence with the Macsyma system for symbolic computation provided the insight that one should avoid complexities of control and concentrate xxv on organizing the data to reﬂect the real structure of the world being modeled. Marvin Minsky and Seymour Papert formed many of our aitudes about programming and its place in our intellectual lives. To them we owe the understanding that computation provides a means of expres- sion for exploring ideas that would otherwise be too complex to deal with precisely. ey emphasize that a student’s ability to write and modify programs provides a powerful medium in which exploring be- comes a natural activity. We also strongly agree with Alan Perlis that programming is lots of fun and we had beer be careful to support the joy of programming. Part of this joy derives from observing great masters at work. We are fortu- nate to have been apprentice programmers at the feet of Bill Gosper and Richard Greenbla. It is diﬃcult to identify all the people who have contributed to the development of our curriculum. We thank all the lecturers, recitation instructors, and tutors who have worked with us over the past ﬁeen years and put in many extra hours on our subject, especially Bill Siebert, Albert Meyer, Joe Stoy, Randy Davis, Louis Braida, Eric Grimson, Rod Brooks, Lynn Stein and Peter Szolovits. We would like to specially ac- knowledge the outstanding teaching contributions of Franklyn Turbak, now at Wellesley; his work in undergraduate instruction set a standard that we can all aspire to. We are grateful to Jerry Saltzer and Jim Miller for helping us grapple with the mysteries of concurrency, and to Peter Szolovits and David McAllester for their contributions to the exposition of nondeterministic evaluation in Chapter 4. Many people have put in signiﬁcant eﬀort presenting this material at other universities. Some of the people we have worked closely with are Jacob Katzenelson at the Technion, Hardy Mayer at the University xxvi of California at Irvine, Joe Stoy at Oxford, Elisha Sacks at Purdue, and Jan Komorowski at the Norwegian University of Science and Technol- ogy. We are exceptionally proud of our colleagues who have received major teaching awards for their adaptations of this subject at other uni- versities, including Kenneth Yip at Yale, Brian Harvey at the University of California at Berkeley, and Dan Huenlocher at Cornell. Al Moyé arranged for us to teach this material to engineers at Hewle- Packard, and for the production of videotapes of these lectures. We would like to thank the talented instructors—in particular Jim Miller, Bill Siebert, and Mike Eisenberg—who have designed continuing edu- cation courses incorporating these tapes and taught them at universities and industry all over the world. Many educators in other countries have put in signiﬁcant work translating the ﬁrst edition. Michel Briand, Pierre Chamard, and An- dré Pic produced a French edition; Susanne Daniels-Herold produced a German edition; and Fumio Motoyoshi produced a Japanese edition. We do not know who produced the Chinese edition, but we consider it an honor to have been selected as the subject of an “unauthorized” translation. It is hard to enumerate all the people who have made technical con- tributions to the development of the Scheme systems we use for in- structional purposes. In addition to Guy Steele, principal wizards have included Chris Hanson, Joe Bowbeer, Jim Miller, Guillermo Rozas, and Stephen Adams. Others who have put in signiﬁcant time are Richard Stallman, Alan Bawden, Kent Pitman, Jon Ta, Neil Mayle, John Lamp- ing, Gwyn Osnos, Tracy Larrabee, George Carree, Soma Chaudhuri, Bill Chiarchiaro, Steven Kirsch, Leigh Klotz, Wayne Noss, Todd Cass, Patrick O’Donnell, Kevin eobald, Daniel Weise, Kenneth Sinclair, An- thony Courtemanche, Henry M. Wu, Andrew Berlin, and Ruth Shyu. xxvii Beyond the implementation, we would like to thank the many people who worked on the Scheme standard, including William Clinger and Jonathan Rees, who edited the R4 RS, and Chris Haynes, David Bartley, Chris Hanson, and Jim Miller, who prepared the standard. Dan Friedman has been a long-time leader of the Scheme commu- nity. e community’s broader work goes beyond issues of language design to encompass signiﬁcant educational innovations, such as the high-school curriculum based on EdScheme by Schemer’s Inc., and the wonderful books by Mike Eisenberg and by Brian Harvey and Mahew Wright. We appreciate the work of those who contributed to making this a real book, especially Terry Ehling, Larry Cohen, and Paul Bethge at the Press. Ella Mazel found the wonderful cover image. For the second edition we are particularly grateful to Bernard and Ella Mazel for help with the book design, and to David Jones, TEX wizard extraordinaire. We also are indebted to those readers who made penetrating comments on the new dra: Jacob Katzenelson, Hardy Mayer, Jim Miller, and es- pecially Brian Harvey, who did unto this book as Julie did unto his book Simply Scheme. Finally, we would like to acknowledge the support of the organiza- tions that have encouraged this work over the years, including support from Hewle-Packard, made possible by Ira Goldstein and Joel Birn- baum, and support from , made possible by Bob Kahn. xxviii Building Abstractions with Procedures e acts of the mind, wherein it exerts its power over simple ideas, are chieﬂy these three: 1. Combining several simple ideas into one compound one, and thus all complex ideas are made. 2. e second is bringing two ideas, whether sim- ple or complex, together, and seing them by one another so as to take a view of them at once, without uniting them into one, by which it gets all its ideas of relations. 3. e third is separating them from all other ideas that accom- pany them in their real existence: this is called abstraction, and thus all its general ideas are made. —John Locke, An Essay Concerning Human Understanding (1690) W the idea of a computational process. Com- putational processes are abstract beings that inhabit computers. As they evolve, processes manipulate other abstract things called data. 1 e evolution of a process is directed by a paern of rules called a pro- gram. People create programs to direct processes. In eﬀect, we conjure the spirits of the computer with our spells. A computational process is indeed much like a sorcerer’s idea of a spirit. It cannot be seen or touched. It is not composed of maer at all. However, it is very real. It can perform intellectual work. It can answer questions. It can aﬀect the world by disbursing money at a bank or by controlling a robot arm in a factory. e programs we use to conjure processes are like a sorcerer’s spells. ey are carefully composed from symbolic expressions in arcane and esoteric programming languages that prescribe the tasks we want our processes to perform. A computational process, in a correctly working computer, executes programs precisely and accurately. us, like the sorcerer’s appren- tice, novice programmers must learn to understand and to anticipate the consequences of their conjuring. Even small errors (usually called bugs or glitches) in programs can have complex and unanticipated con- sequences. Fortunately, learning to program is considerably less dangerous than learning sorcery, because the spirits we deal with are conveniently con- tained in a secure way. Real-world programming, however, requires care, expertise, and wisdom. A small bug in a computer-aided design program, for example, can lead to the catastrophic collapse of an air- plane or a dam or the self-destruction of an industrial robot. Master soware engineers have the ability to organize programs so that they can be reasonably sure that the resulting processes will per- form the tasks intended. ey can visualize the behavior of their sys- tems in advance. ey know how to structure programs so that unan- ticipated problems do not lead to catastrophic consequences, and when problems do arise, they can debug their programs. Well-designed com- 2 putational systems, like well-designed automobiles or nuclear reactors, are designed in a modular manner, so that the parts can be constructed, replaced, and debugged separately. Programming in Lisp We need an appropriate language for describing processes, and we will use for this purpose the programming language Lisp. Just as our every- day thoughts are usually expressed in our natural language (such as En- glish, French, or Japanese), and descriptions of quantitative phenomena are expressed with mathematical notations, our procedural thoughts will be expressed in Lisp. Lisp was invented in the late 1950s as a for- malism for reasoning about the use of certain kinds of logical expres- sions, called recursion equations, as a model for computation. e lan- guage was conceived by John McCarthy and is based on his paper “Re- cursive Functions of Symbolic Expressions and eir Computation by Machine” (McCarthy 1960). Despite its inception as a mathematical formalism, Lisp is a practi- cal programming language. A Lisp interpreter is a machine that carries out processes described in the Lisp language. e ﬁrst Lisp interpreter was implemented by McCarthy with the help of colleagues and stu- dents in the Artiﬁcial Intelligence Group of the Research Laboratory of Electronics and in the Computation Center.1 Lisp, whose name is an acronym for LISt Processing, was designed to provide symbol- manipulating capabilities for aacking programming problems such as the symbolic diﬀerentiation and integration of algebraic expressions. It included for this purpose new data objects known as atoms and lists, 1 e Lisp 1 Programmer’s Manual appeared in 1960, and the Lisp 1.5 Programmer’s Manual (McCarthy et al. 1965) was published in 1962. e early history of Lisp is de- scribed in McCarthy 1978. 3 which most strikingly set it apart from all other languages of the period. Lisp was not the product of a concerted design eﬀort. Instead, it evolved informally in an experimental manner in response to users’ needs and to pragmatic implementation considerations. Lisp’s informal evolution has continued through the years, and the community of Lisp users has traditionally resisted aempts to promulgate any “oﬃcial” deﬁnition of the language. is evolution, together with the ﬂexibility and elegance of the initial conception, has enabled Lisp, which is the sec- ond oldest language in widespread use today (only Fortran is older), to continually adapt to encompass the most modern ideas about program design. us, Lisp is by now a family of dialects, which, while sharing most of the original features, may diﬀer from one another in signiﬁcant ways. e dialect of Lisp used in this book is called Scheme.2 Because of its experimental character and its emphasis on symbol manipulation, Lisp was at ﬁrst very ineﬃcient for numerical compu- tations, at least in comparison with Fortran. Over the years, however, 2 e two dialects in which most major Lisp programs of the 1970s were wrien are MacLisp (Moon 1978; Pitman 1983), developed at the Project , and Interlisp (Teitelman 1974), developed at Bolt Beranek and Newman Inc. and the Xerox Palo Alto Research Center. Portable Standard Lisp (Hearn 1969; Griss 1981) was a Lisp dialect designed to be easily portable between diﬀerent machines. MacLisp spawned a number of subdialects, such as Franz Lisp, which was developed at the University of California at Berkeley, and Zetalisp (Moon and Weinreb 1981), which was based on a special- purpose processor designed at the Artiﬁcial Intelligence Laboratory to run Lisp very eﬃciently. e Lisp dialect used in this book, called Scheme (Steele and Sussman 1975), was invented in 1975 by Guy Lewis Steele Jr. and Gerald Jay Sussman of the Artiﬁcial Intelligence Laboratory and later reimplemented for instructional use at . Scheme became an standard in 1990 (IEEE 1990). e Common Lisp dialect (Steele 1982, Steele 1990) was developed by the Lisp community to combine features from the earlier Lisp dialects to make an industrial standard for Lisp. Common Lisp became an standard in 1994 (ANSI 1994). 4 Lisp compilers have been developed that translate programs into ma- chine code that can perform numerical computations reasonably eﬃ- ciently. And for special applications, Lisp has been used with great ef- fectiveness.3 Although Lisp has not yet overcome its old reputation as hopelessly ineﬃcient, Lisp is now used in many applications where ef- ﬁciency is not the central concern. For example, Lisp has become a lan- guage of choice for operating-system shell languages and for extension languages for editors and computer-aided design systems. If Lisp is not a mainstream language, why are we using it as the framework for our discussion of programming? Because the language possesses unique features that make it an excellent medium for studying important programming constructs and data structures and for relating them to the linguistic features that support them. e most signiﬁcant of these features is the fact that Lisp descriptions of processes, called proce- dures, can themselves be represented and manipulated as Lisp data. e importance of this is that there are powerful program-design techniques that rely on the ability to blur the traditional distinction between “pas- sive” data and “active” processes. As we shall discover, Lisp’s ﬂexibility in handling procedures as data makes it one of the most convenient languages in existence for exploring these techniques. e ability to represent procedures as data also makes Lisp an excellent language for writing programs that must manipulate other programs as data, such as the interpreters and compilers that support computer languages. Above and beyond these considerations, programming in Lisp is great fun. 3 One such special application was a breakthrough computation of scientiﬁc importance—an integration of the motion of the Solar System that extended previous results by nearly two orders of magnitude, and demonstrated that the dynamics of the Solar System is chaotic. is computation was made possible by new integration al- gorithms, a special-purpose compiler, and a special-purpose computer all implemented with the aid of soware tools wrien in Lisp (Abelson et al. 1992; Sussman and Wisdom 1992). 5 1.1 The Elements of Programming A powerful programming language is more than just a means for in- structing a computer to perform tasks. e language also serves as a framework within which we organize our ideas about processes. us, when we describe a language, we should pay particular aention to the means that the language provides for combining simple ideas to form more complex ideas. Every powerful language has three mechanisms for accomplishing this: • primitive expressions, which represent the simplest entities the language is concerned with, • means of combination, by which compound elements are built from simpler ones, and • means of abstraction, by which compound elements can be named and manipulated as units. In programming, we deal with two kinds of elements: procedures and data. (Later we will discover that they are really not so distinct.) Infor- mally, data is “stuﬀ” that we want to manipulate, and procedures are descriptions of the rules for manipulating the data. us, any powerful programming language should be able to describe primitive data and primitive procedures and should have methods for combining and ab- stracting procedures and data. In this chapter we will deal only with simple numerical data so that we can focus on the rules for building procedures.4 In later chapters we 4 e characterization of numbers as “simple data” is a barefaced bluﬀ. In fact, the treatment of numbers is one of the trickiest and most confusing aspects of any pro- 6 will see that these same rules allow us to build procedures to manipulate compound data as well. 1.1.1 Expressions One easy way to get started at programming is to examine some typical interactions with an interpreter for the Scheme dialect of Lisp. Imagine that you are siing at a computer terminal. You type an expression, and the interpreter responds by displaying the result of its evaluating that expression. One kind of primitive expression you might type is a number. (More precisely, the expression that you type consists of the numerals that represent the number in base 10.) If you present Lisp with a number 486 the interpreter will respond by printing5 486 gramming language. Some typical issues involved are these: Some computer systems distinguish integers, such as 2, from real numbers, such as 2.71. Is the real number 2.00 diﬀerent from the integer 2? Are the arithmetic operations used for integers the same as the operations used for real numbers? Does 6 divided by 2 produce 3, or 3.0? How large a number can we represent? How many decimal places of accuracy can we repre- sent? Is the range of integers the same as the range of real numbers? Above and beyond these questions, of course, lies a collection of issues concerning roundoﬀ and trunca- tion errors—the entire science of numerical analysis. Since our focus in this book is on large-scale program design rather than on numerical techniques, we are going to ignore these problems. e numerical examples in this chapter will exhibit the usual roundoﬀ behavior that one observes when using arithmetic operations that preserve a limited number of decimal places of accuracy in noninteger operations. 5 roughout this book, when we wish to emphasize the distinction between the input typed by the user and the response printed by the interpreter, we will show the laer in slanted characters. 7 Expressions representing numbers may be combined with an expres- sion representing a primitive procedure (such as + or *) to form a com- pound expression that represents the application of the procedure to those numbers. For example: (+ 137 349) 486 (- 1000 334) 666 (* 5 99) 495 (/ 10 5) 2 (+ 2.7 10) 12.7 Expressions such as these, formed by delimiting a list of expressions within parentheses in order to denote procedure application, are called combinations. e lemost element in the list is called the operator, and the other elements are called operands. e value of a combination is obtained by applying the procedure speciﬁed by the operator to the ar- guments that are the values of the operands. e convention of placing the operator to the le of the operands is known as preﬁx notation, and it may be somewhat confusing at ﬁrst because it departs signiﬁcantly from the customary mathematical con- vention. Preﬁx notation has several advantages, however. One of them is that it can accommodate procedures that may take an arbitrary num- ber of arguments, as in the following examples: 8 (+ 21 35 12 7) 75 (* 25 4 12) 1200 No ambiguity can arise, because the operator is always the lemost el- ement and the entire combination is delimited by the parentheses. A second advantage of preﬁx notation is that it extends in a straight- forward way to allow combinations to be nested, that is, to have combi- nations whose elements are themselves combinations: (+ (* 3 5) (- 10 6)) 19 ere is no limit (in principle) to the depth of such nesting and to the overall complexity of the expressions that the Lisp interpreter can eval- uate. It is we humans who get confused by still relatively simple expres- sions such as (+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6)) which the interpreter would readily evaluate to be 57. We can help our- selves by writing such an expression in the form (+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6)) following a formaing convention known as prey-printing, in which each long combination is wrien so that the operands are aligned ver- tically. e resulting indentations display clearly the structure of the 9 expression.6 Even with complex expressions, the interpreter always operates in the same basic cycle: It reads an expression from the terminal, evaluates the expression, and prints the result. is mode of operation is oen expressed by saying that the interpreter runs in a read-eval-print loop. Observe in particular that it is not necessary to explicitly instruct the interpreter to print the value of the expression.7 1.1.2 Naming and the Environment A critical aspect of a programming language is the means it provides for using names to refer to computational objects. We say that the name identiﬁes a variable whose value is the object. In the Scheme dialect of Lisp, we name things with define. Typing (define size 2) causes the interpreter to associate the value 2 with the name size.8 Once the name size has been associated with the number 2, we can refer to the value 2 by name: size 2 6 Lisp systems typically provide features to aid the user in formaing expressions. Two especially useful features are one that automatically indents to the proper prey- print position whenever a new line is started and one that highlights the matching le parenthesis whenever a right parenthesis is typed. 7 Lisp obeys the convention that every expression has a value. is convention, to- gether with the old reputation of Lisp as an ineﬃcient language, is the source of the quip by Alan Perlis (paraphrasing Oscar Wilde) that “Lisp programmers know the value of everything but the cost of nothing.” 8 In this book, we do not show the interpreter’s response to evaluating deﬁnitions, since this is highly implementation-dependent. 10 (* 5 size) 10 Here are further examples of the use of define: (define pi 3.14159) (define radius 10) (* pi (* radius radius)) 314.159 (define circumference (* 2 pi radius)) circumference 62.8318 define is our language’s simplest means of abstraction, for it allows us to use simple names to refer to the results of compound operations, such as the circumference computed above. In general, computational objects may have very complex structures, and it would be extremely inconvenient to have to remember and repeat their details each time we want to use them. Indeed, complex programs are constructed by build- ing, step by step, computational objects of increasing complexity. e interpreter makes this step-by-step program construction particularly convenient because name-object associations can be created incremen- tally in successive interactions. is feature encourages the incremental development and testing of programs and is largely responsible for the fact that a Lisp program usually consists of a large number of relatively simple procedures. It should be clear that the possibility of associating values with sym- bols and later retrieving them means that the interpreter must maintain some sort of memory that keeps track of the name-object pairs. is memory is called the environment (more precisely the global environ- ment, since we will see later that a computation may involve a number 11 of diﬀerent environments).9 1.1.3 Evaluating Combinations One of our goals in this chapter is to isolate issues about thinking pro- cedurally. As a case in point, let us consider that, in evaluating combi- nations, the interpreter is itself following a procedure. To evaluate a combination, do the following: 1. Evaluate the subexpressions of the combination. 2. Apply the procedure that is the value of the lemost subexpres- sion (the operator) to the arguments that are the values of the other subexpressions (the operands). Even this simple rule illustrates some important points about processes in general. First, observe that the ﬁrst step dictates that in order to ac- complish the evaluation process for a combination we must ﬁrst per- form the evaluation process on each element of the combination. us, the evaluation rule is recursive in nature; that is, it includes, as one of its steps, the need to invoke the rule itself.10 Notice how succinctly the idea of recursion can be used to express what, in the case of a deeply nested combination, would otherwise be viewed as a rather complicated process. For example, evaluating 9 Chapter 3 will show that this notion of environment is crucial, both for under- standing how the interpreter works and for implementing interpreters. 10 It may seem strange that the evaluation rule says, as part of the ﬁrst step, that we should evaluate the lemost element of a combination, since at this point that can only be an operator such as + or * representing a built-in primitive procedure such as addition or multiplication. We will see later that it is useful to be able to work with combinations whose operators are themselves compound expressions. 12 390 * 26 15 + 2 24 + 3 5 7 * 4 6 Figure 1.1: Tree representation, showing the value of each subcombination. (* (+ 2 (* 4 6)) (+ 3 5 7)) requires that the evaluation rule be applied to four diﬀerent combina- tions. We can obtain a picture of this process by representing the combi- nation in the form of a tree, as shown in Figure 1.1. Each combination is represented by a node with branches corresponding to the operator and the operands of the combination stemming from it. e terminal nodes (that is, nodes with no branches stemming from them) represent either operators or numbers. Viewing evaluation in terms of the tree, we can imagine that the values of the operands percolate upward, starting from the terminal nodes and then combining at higher and higher levels. In general, we shall see that recursion is a very powerful technique for dealing with hierarchical, treelike objects. In fact, the “percolate values upward” form of the evaluation rule is an example of a general kind of process known as tree accumulation. Next, observe that the repeated application of the ﬁrst step brings us to the point where we need to evaluate, not combinations, but primitive expressions such as numerals, built-in operators, or other names. We 13 take care of the primitive cases by stipulating that • the values of numerals are the numbers that they name, • the values of built-in operators are the machine instruction se- quences that carry out the corresponding operations, and • the values of other names are the objects associated with those names in the environment. We may regard the second rule as a special case of the third one by stip- ulating that symbols such as + and * are also included in the global envi- ronment, and are associated with the sequences of machine instructions that are their “values.” e key point to notice is the role of the environ- ment in determining the meaning of the symbols in expressions. In an interactive language such as Lisp, it is meaningless to speak of the value of an expression such as (+ x 1) without specifying any information about the environment that would provide a meaning for the symbol x (or even for the symbol +). As we shall see in Chapter 3, the general notion of the environment as providing a context in which evaluation takes place will play an important role in our understanding of program execution. Notice that the evaluation rule given above does not handle deﬁni- tions. For instance, evaluating (define x 3) does not apply define to two arguments, one of which is the value of the symbol x and the other of which is 3, since the purpose of the define is precisely to associate x with a value. (at is, (define x 3) is not a combination.) Such exceptions to the general evaluation rule are called special forms. define is the only example of a special form that we have seen so far, but we will meet others shortly. Each special form has its own evalu- ation rule. e various kinds of expressions (each with its associated 14 evaluation rule) constitute the syntax of the programming language. In comparison with most other programming languages, Lisp has a very simple syntax; that is, the evaluation rule for expressions can be de- scribed by a simple general rule together with specialized rules for a small number of special forms.11 1.1.4 Compound Procedures We have identiﬁed in Lisp some of the elements that must appear in any powerful programming language: • Numbers and arithmetic operations are primitive data and proce- dures. • Nesting of combinations provides a means of combining opera- tions. • Deﬁnitions that associate names with values provide a limited means of abstraction. Now we will learn about procedure deﬁnitions, a much more powerful abstraction technique by which a compound operation can be given a name and then referred to as a unit. 11 Special syntactic forms that are simply convenient alternative surface structures for things that can be wrien in more uniform ways are sometimes called syntactic sugar, to use a phrase coined by Peter Landin. In comparison with users of other lan- guages, Lisp programmers, as a rule, are less concerned with maers of syntax. (By contrast, examine any Pascal manual and notice how much of it is devoted to descrip- tions of syntax.) is disdain for syntax is due partly to the ﬂexibility of Lisp, which makes it easy to change surface syntax, and partly to the observation that many “con- venient” syntactic constructs, which make the language less uniform, end up causing more trouble than they are worth when programs become large and complex. In the words of Alan Perlis, “Syntactic sugar causes cancer of the semicolon.” 15 We begin by examining how to express the idea of “squaring.” We might say, “To square something, multiply it by itself.” is is expressed in our language as (define (square x) (* x x)) We can understand this in the following way: (define (square x) (* x x)) | | | | | | To square something, multiply it by itself. We have here a compound procedure, which has been given the name square. e procedure represents the operation of multiplying some- thing by itself. e thing to be multiplied is given a local name, x, which plays the same role that a pronoun plays in natural language. Evaluating the deﬁnition creates this compound procedure and associates it with the name square.12 e general form of a procedure deﬁnition is (define (⟨name⟩ ⟨formal parameters⟩) ⟨ body⟩) e ⟨name ⟩ is a symbol to be associated with the procedure deﬁnition in the environment.13 e ⟨formal parameters ⟩ are the names used within the body of the procedure to refer to the corresponding arguments of the procedure. e ⟨body ⟩ is an expression that will yield the value of 12 Observe that there are two diﬀerent operations being combined here: we are creat- ing the procedure, and we are giving it the name square. It is possible, indeed important, to be able to separate these two notions—to create procedures without naming them, and to give names to procedures that have already been created. We will see how to do this in Section 1.3.2. 13 roughout this book, we will describe the general syntax of expressions by using italic symbols delimited by angle brackets—e.g., ⟨name ⟩—to denote the “slots” in the expression to be ﬁlled in when such an expression is actually used. 16 the procedure application when the formal parameters are replaced by the actual arguments to which the procedure is applied.14 e ⟨name ⟩ and the ⟨formal parameters ⟩ are grouped within parentheses, just as they would be in an actual call to the procedure being deﬁned. Having deﬁned square, we can now use it: (square 21) 441 (square (+ 2 5)) 49 (square (square 3)) 81 We can also use square as a building block in deﬁning other procedures. For example, x 2 + y 2 can be expressed as (+ (square x) (square y)) We can easily deﬁne a procedure sum-of-squares that, given any two numbers as arguments, produces the sum of their squares: (define (sum-of-squares x y) (+ (square x) (square y))) (sum-of-squares 3 4) 25 Now we can use sum-of-squares as a building block in constructing further procedures: (define (f a) (sum-of-squares (+ a 1) (* a 2))) (f 5) 136 14 More generally, the body of the procedure can be a sequence of expressions. In this case, the interpreter evaluates each expression in the sequence in turn and returns the value of the ﬁnal expression as the value of the procedure application. 17 Compound procedures are used in exactly the same way as primitive procedures. Indeed, one could not tell by looking at the deﬁnition of sum-of-squares given above whether square was built into the inter- preter, like + and *, or deﬁned as a compound procedure. 1.1.5 The Substitution Model for Procedure Application To evaluate a combination whose operator names a compound proce- dure, the interpreter follows much the same process as for combina- tions whose operators name primitive procedures, which we described in Section 1.1.3. at is, the interpreter evaluates the elements of the combination and applies the procedure (which is the value of the oper- ator of the combination) to the arguments (which are the values of the operands of the combination). We can assume that the mechanism for applying primitive proce- dures to arguments is built into the interpreter. For compound proce- dures, the application process is as follows: To apply a compound procedure to arguments, evaluate the body of the procedure with each formal parameter replaced by the corresponding argument. To illustrate this process, let’s evaluate the combination (f 5) where f is the procedure deﬁned in Section 1.1.4. We begin by retrieving the body of f: (sum-of-squares (+ a 1) (* a 2)) en we replace the formal parameter a by the argument 5: (sum-of-squares (+ 5 1) (* 5 2)) 18 us the problem reduces to the evaluation of a combination with two operands and an operator sum-of-squares. Evaluating this combina- tion involves three subproblems. We must evaluate the operator to get the procedure to be applied, and we must evaluate the operands to get the arguments. Now (+ 5 1) produces 6 and (* 5 2) produces 10, so we must apply the sum-of-squares procedure to 6 and 10. ese values are substituted for the formal parameters x and y in the body of sum- of-squares, reducing the expression to (+ (square 6) (square 10)) If we use the deﬁnition of square, this reduces to (+ (* 6 6) (* 10 10)) which reduces by multiplication to (+ 36 100) and ﬁnally to 136 e process we have just described is called the substitution model for procedure application. It can be taken as a model that determines the “meaning” of procedure application, insofar as the procedures in this chapter are concerned. However, there are two points that should be stressed: • e purpose of the substitution is to help us think about proce- dure application, not to provide a description of how the inter- preter really works. Typical interpreters do not evaluate proce- dure applications by manipulating the text of a procedure to sub- stitute values for the formal parameters. In practice, the “substi- tution” is accomplished by using a local environment for the for- mal parameters. We will discuss this more fully in Chapter 3 and 19 Chapter 4 when we examine the implementation of an interpreter in detail. • Over the course of this book, we will present a sequence of in- creasingly elaborate models of how interpreters work, culminat- ing with a complete implementation of an interpreter and com- piler in Chapter 5. e substitution model is only the ﬁrst of these models—a way to get started thinking formally about the evalu- ation process. In general, when modeling phenomena in science and engineering, we begin with simpliﬁed, incomplete models. As we examine things in greater detail, these simple models be- come inadequate and must be replaced by more reﬁned models. e substitution model is no exception. In particular, when we address in Chapter 3 the use of procedures with “mutable data,” we will see that the substitution model breaks down and must be replaced by a more complicated model of procedure application.15 Applicative order versus normal order According to the description of evaluation given in Section 1.1.3, the interpreter ﬁrst evaluates the operator and operands and then applies the resulting procedure to the resulting arguments. is is not the only way to perform evaluation. An alternative evaluation model would not evaluate the operands until their values were needed. Instead it would 15 Despite the simplicity of the substitution idea, it turns out to be surprisingly com- plicated to give a rigorous mathematical deﬁnition of the substitution process. e problem arises from the possibility of confusion between the names used for the formal parameters of a procedure and the (possibly identical) names used in the expressions to which the procedure may be applied. Indeed, there is a long history of erroneous def- initions of substitution in the literature of logic and programming semantics. See Stoy 1977 for a careful discussion of substitution. 20 ﬁrst substitute operand expressions for parameters until it obtained an expression involving only primitive operators, and would then perform the evaluation. If we used this method, the evaluation of (f 5) would proceed according to the sequence of expansions (sum-of-squares (+ 5 1) (* 5 2)) (+ (square (+ 5 1)) (square (* 5 2)) ) (+ (* (+ 5 1) (+ 5 1)) (* (* 5 2) (* 5 2))) followed by the reductions (+ (* 6 6) (* 10 10)) (+ 36 100) 136 is gives the same answer as our previous evaluation model, but the process is diﬀerent. In particular, the evaluations of (+ 5 1) and (* 5 2) are each performed twice here, corresponding to the reduction of the expression (* x x) with x replaced respectively by (+ 5 1) and (* 5 2). is alternative “fully expand and then reduce” evaluation method is known as normal-order evaluation, in contrast to the “evaluate the arguments and then apply” method that the interpreter actually uses, which is called applicative-order evaluation. It can be shown that, for procedure applications that can be modeled using substitution (includ- ing all the procedures in the ﬁrst two chapters of this book) and that yield legitimate values, normal-order and applicative-order evaluation produce the same value. (See Exercise 1.5 for an instance of an “illegit- imate” value where normal-order and applicative-order evaluation do not give the same result.) Lisp uses applicative-order evaluation, partly because of the addi- tional eﬃciency obtained from avoiding multiple evaluations of expres- sions such as those illustrated with (+ 5 1) and (* 5 2) above and, more 21 signiﬁcantly, because normal-order evaluation becomes much more com- plicated to deal with when we leave the realm of procedures that can be modeled by substitution. On the other hand, normal-order evaluation can be an extremely valuable tool, and we will investigate some of its implications in Chapter 3 and Chapter 4.16 1.1.6 Conditional Expressions and Predicates e expressive power of the class of procedures that we can deﬁne at this point is very limited, because we have no way to make tests and to perform diﬀerent operations depending on the result of a test. For instance, we cannot deﬁne a procedure that computes the absolute value of a number by testing whether the number is positive, negative, or zero and taking diﬀerent actions in the diﬀerent cases according to the rule x if x > 0, |x | = 0 if x = 0, −x if x < 0. is construct is called a case analysis, and there is a special form in Lisp for notating such a case analysis. It is called cond (which stands for “conditional”), and it is used as follows: (define (abs x) (cond ((> x 0) x) ((= x 0) 0) ((< x 0) (- x)))) e general form of a conditional expression is 16 In Chapter 3 we will introduce stream processing, which is a way of handling appar- ently “inﬁnite” data structures by incorporating a limited form of normal-order evalu- ation. In Section 4.2 we will modify the Scheme interpreter to produce a normal-order variant of Scheme. 22 (cond (⟨p1 ⟩ ⟨e1 ⟩) (⟨p2 ⟩ ⟨e2 ⟩) ... (⟨pn ⟩ ⟨en ⟩)) consisting of the symbol cond followed by parenthesized pairs of ex- pressions (⟨p⟩ ⟨e⟩) called clauses. e ﬁrst expression in each pair is a predicate—that is, an expression whose value is interpreted as either true or false.17 Conditional expressions are evaluated as follows. e predicate ⟨p 1 ⟩ is evaluated ﬁrst. If its value is false, then ⟨p 2 ⟩ is evaluated. If ⟨p 2 ⟩’s value is also false, then ⟨p 3 ⟩ is evaluated. is process continues until a predicate is found whose value is true, in which case the interpreter returns the value of the corresponding consequent expression ⟨e⟩ of the clause as the value of the conditional expression. If none of the ⟨p⟩’s is found to be true, the value of the cond is undeﬁned. e word predicate is used for procedures that return true or false, as well as for expressions that evaluate to true or false. e absolute- value procedure abs makes use of the primitive predicates >, <, and =.18 ese take two numbers as arguments and test whether the ﬁrst number is, respectively, greater than, less than, or equal to the second number, returning true or false accordingly. Another way to write the absolute-value procedure is 17 “Interpreted as either true or false” means this: In Scheme, there are two distin- guished values that are denoted by the constants #t and #f. When the interpreter checks a predicate’s value, it interprets #f as false. Any other value is treated as true. (us, providing #t is logically unnecessary, but it is convenient.) In this book we will use names true and false, which are associated with the values #t and #f respectively. 18 abs also uses the “minus” operator -, which, when used with a single operand, as in (- x), indicates negation. 23 (define (abs x) (cond ((< x 0) (- x)) (else x))) which could be expressed in English as “If x is less than zero return −x; otherwise return x.” else is a special symbol that can be used in place of the ⟨p⟩ in the ﬁnal clause of a cond. is causes the cond to return as its value the value of the corresponding ⟨e⟩ whenever all previous clauses have been bypassed. In fact, any expression that always evaluates to a true value could be used as the ⟨p⟩ here. Here is yet another way to write the absolute-value procedure: (define (abs x) (if (< x 0) (- x) x)) is uses the special form if, a restricted type of conditional that can be used when there are precisely two cases in the case analysis. e general form of an if expression is (if ⟨ predicate ⟩ ⟨ consequent ⟩ ⟨ alternative ⟩) To evaluate an if expression, the interpreter starts by evaluating the ⟨predicate ⟩ part of the expression. If the ⟨predicate ⟩ evaluates to a true value, the interpreter then evaluates the ⟨consequent ⟩ and returns its value. Otherwise it evaluates the ⟨alternative ⟩ and returns its value.19 In addition to primitive predicates such as <, =, and >, there are log- ical composition operations, which enable us to construct compound 19 A minor diﬀerence between if and cond is that the ⟨e⟩ part of each cond clause may be a sequence of expressions. If the corresponding ⟨p⟩ is found to be true, the expres- sions ⟨e⟩ are evaluated in sequence and the value of the ﬁnal expression in the sequence is returned as the value of the cond. In an if expression, however, the ⟨consequent ⟩ and ⟨alternative ⟩ must be single expressions. 24 predicates. e three most frequently used are these: • (and ⟨e 1 ⟩ . . . ⟨en ⟩) e interpreter evaluates the expressions ⟨e ⟩ one at a time, in le- to-right order. If any ⟨e ⟩ evaluates to false, the value of the and expression is false, and the rest of the ⟨e ⟩’s are not evaluated. If all ⟨e ⟩’s evaluate to true values, the value of the and expression is the value of the last one. • (or ⟨e 1 ⟩ . . . ⟨en ⟩) e interpreter evaluates the expressions ⟨e ⟩ one at a time, in le- to-right order. If any ⟨e ⟩ evaluates to a true value, that value is returned as the value of the or expression, and the rest of the ⟨e ⟩’s are not evaluated. If all ⟨e ⟩’s evaluate to false, the value of the or expression is false. • (not ⟨e⟩) e value of a not expression is true when the expression ⟨e ⟩ evaluates to false, and false otherwise. Notice that and and or are special forms, not procedures, because the subexpressions are not necessarily all evaluated. not is an ordinary pro- cedure. As an example of how these are used, the condition that a number x be in the range 5 < x < 10 may be expressed as (and (> x 5) (< x 10)) As another example, we can deﬁne a predicate to test whether one num- ber is greater than or equal to another as (define (>= x y) (or (> x y) (= x y))) 25 or alternatively as (define (>= x y) (not (< x y))) Exercise 1.1: Below is a sequence of expressions. What is the result printed by the interpreter in response to each ex- pression? Assume that the sequence is to be evaluated in the order in which it is presented. 10 (+ 5 3 4) (- 9 1) (/ 6 2) (+ (* 2 4) (- 4 6)) (define a 3) (define b (+ a 1)) (+ a b (* a b)) (= a b) (if (and (> b a) (< b (* a b))) b a) (cond ((= a 4) 6) ((= b 4) (+ 6 7 a)) (else 25)) (+ 2 (if (> b a) b a)) (* (cond ((> a b) a) ((< a b) b) (else -1)) (+ a 1)) 26 Exercise 1.2: Translate the following expression into preﬁx form: 5 + 4 + (2 − (3 − (6 + 54 ))) . 3(6 − 2)(2 − 7) Exercise 1.3: Deﬁne a procedure that takes three numbers as arguments and returns the sum of the squares of the two larger numbers. Exercise 1.4: Observe that our model of evaluation allows for combinations whose operators are compound expres- sions. Use this observation to describe the behavior of the following procedure: (define (a-plus-abs-b a b) ((if (> b 0) + -) a b)) Exercise 1.5: Ben Bitdiddle has invented a test to determine whether the interpreter he is faced with is using applicative- order evaluation or normal-order evaluation. He deﬁnes the following two procedures: (define (p) (p)) (define (test x y) (if (= x 0) 0 y)) en he evaluates the expression (test 0 (p)) What behavior will Ben observe with an interpreter that uses applicative-order evaluation? What behavior will he observe with an interpreter that uses normal-order evalu- ation? Explain your answer. (Assume that the evaluation 27 rule for the special form if is the same whether the in- terpreter is using normal or applicative order: e predi- cate expression is evaluated ﬁrst, and the result determines whether to evaluate the consequent or the alternative ex- pression.) 1.1.7 Example: Square Roots by Newton’s Method Procedures, as introduced above, are much like ordinary mathematical functions. ey specify a value that is determined by one or more pa- rameters. But there is an important diﬀerence between mathematical functions and computer procedures. Procedures must be eﬀective. As a case in point, consider the problem of computing square roots. We can deﬁne the square-root function as √ x = the y such that y ≥ 0 and y 2 = x . is describes a perfectly legitimate mathematical function. We could use it to recognize whether one number is the square root of another, or to derive facts about square roots in general. On the other hand, the deﬁnition does not describe a procedure. Indeed, it tells us almost noth- ing about how to actually ﬁnd the square root of a given number. It will not help maers to rephrase this deﬁnition in pseudo-Lisp: (define (sqrt x) (the y (and (>= y 0) (= (square y) x)))) is only begs the question. e contrast between function and procedure is a reﬂection of the general distinction between describing properties of things and describ- ing how to do things, or, as it is sometimes referred to, the distinction 28 between declarative knowledge and imperative knowledge. In mathe- matics we are usually concerned with declarative (what is) descriptions, whereas in computer science we are usually concerned with imperative (how to) descriptions.20 How does one compute square roots? e most common way is to use Newton’s method of successive approximations, which says that whenever we have a guess y for the value of the square root of a number x , we can perform a simple manipulation to get a beer guess (one closer to the actual square root) by averaging y with x/y.21 For example, we can compute the square root of 2 as follows. Suppose our initial guess is 1: Guess Quotient Average 1 (2/1) = 2 ((2 + 1)/2) = 1.5 1.5 (2/1.5) = 1.3333 ((1.3333 + 1.5)/2) = 1.4167 1.4167 (2/1.4167) = 1.4118 ((1.4167 + 1.4118)/2) = 1.4142 1.4142 ... ... 20 Declarative and imperative descriptions are intimately related, as indeed are math- ematics and computer science. For instance, to say that the answer produced by a pro- gram is “correct” is to make a declarative statement about the program. ere is a large amount of research aimed at establishing techniques for proving that programs are correct, and much of the technical diﬃculty of this subject has to do with negotiating the transition between imperative statements (from which programs are constructed) and declarative statements (which can be used to deduce things). In a related vein, an important current area in programming-language design is the exploration of so-called very high-level languages, in which one actually programs in terms of declarative state- ments. e idea is to make interpreters sophisticated enough so that, given “what is” knowledge speciﬁed by the programmer, they can generate “how to” knowledge auto- matically. is cannot be done in general, but there are important areas where progress has been made. We shall revisit this idea in Chapter 4. 21 is square-root algorithm is actually a special case of Newton’s method, which is a general technique for ﬁnding roots of equations. e square-root algorithm itself was developed by Heron of Alexandria in the ﬁrst century .. We will see how to express the general Newton’s method as a Lisp procedure in Section 1.3.4. 29 Continuing this process, we obtain beer and beer approximations to the square root. Now let’s formalize the process in terms of procedures. We start with a value for the radicand (the number whose square root we are trying to compute) and a value for the guess. If the guess is good enough for our purposes, we are done; if not, we must repeat the process with an improved guess. We write this basic strategy as a procedure: (define (sqrt-iter guess x) (if (good-enough? guess x) guess (sqrt-iter (improve guess x) x))) A guess is improved by averaging it with the quotient of the radicand and the old guess: (define (improve guess x) (average guess (/ x guess))) where (define (average x y) (/ (+ x y) 2)) We also have to say what we mean by “good enough.” e following will do for illustration, but it is not really a very good test. (See Exercise 1.7.) e idea is to improve the answer until it is close enough so that its square diﬀers from the radicand by less than a predetermined tolerance (here 0.001):22 (define (good-enough? guess x) (< (abs (- (square guess) x)) 0.001)) 22 We will usually give predicates names ending with question marks, to help us re- member that they are predicates. is is just a stylistic convention. As far as the inter- preter is concerned, the question mark is just an ordinary character. 30 Finally, we need a way to get started. For instance, we can always guess that the square root of any number is 1:23 (define (sqrt x) (sqrt-iter 1.0 x)) If we type these deﬁnitions to the interpreter, we can use sqrt just as we can use any procedure: (sqrt 9) 3.00009155413138 (sqrt (+ 100 37)) 11.704699917758145 (sqrt (+ (sqrt 2) (sqrt 3))) 1.7739279023207892 (square (sqrt 1000)) 1000.000369924366 e sqrt program also illustrates that the simple procedural language we have introduced so far is suﬃcient for writing any purely numeri- cal program that one could write in, say, C or Pascal. is might seem surprising, since we have not included in our language any iterative 23 Observe that we express our initial guess as 1.0 rather than 1. is would not make any diﬀerence in many Lisp implementations. Scheme, however, distinguishes be- tween exact integers and decimal values, and dividing two integers produces a rational number rather than a decimal. For example, dividing 10 by 6 yields 5/3, while dividing 10.0 by 6.0 yields 1.6666666666666667. (We will learn how to implement arithmetic on rational numbers in Section 2.1.1.) If we start with an initial guess of 1 in our square-root program, and x is an exact integer, all subsequent values produced in the square-root computation will be rational numbers rather than decimals. Mixed operations on ratio- nal numbers and decimals always yield decimals, so starting with an initial guess of 1.0 forces all subsequent values to be decimals. 31 (looping) constructs that direct the computer to do something over and over again. sqrt-iter, on the other hand, demonstrates how iteration can be accomplished using no special construct other than the ordinary ability to call a procedure.24 Exercise 1.6: Alyssa P. Hacker doesn’t see why if needs to be provided as a special form. “Why can’t I just deﬁne it as an ordinary procedure in terms of cond?” she asks. Alyssa’s friend Eva Lu Ator claims this can indeed be done, and she deﬁnes a new version of if: (define (new-if predicate then-clause else-clause) (cond (predicate then-clause) (else else-clause))) Eva demonstrates the program for Alyssa: (new-if (= 2 3) 0 5) 5 (new-if (= 1 1) 0 5) 0 Delighted, Alyssa uses new-if to rewrite the square-root program: (define (sqrt-iter guess x) (new-if (good-enough? guess x) guess (sqrt-iter (improve guess x) x))) What happens when Alyssa aempts to use this to compute square roots? Explain. 24 Readers who are worried about the eﬃciency issues involved in using procedure calls to implement iteration should note the remarks on “tail recursion” in Section 1.2.1. 32 Exercise 1.7: e good-enough? test used in computing square roots will not be very eﬀective for ﬁnding the square roots of very small numbers. Also, in real computers, arith- metic operations are almost always performed with lim- ited precision. is makes our test inadequate for very large numbers. Explain these statements, with examples showing how the test fails for small and large numbers. An alterna- tive strategy for implementing good-enough? is to watch how guess changes from one iteration to the next and to stop when the change is a very small fraction of the guess. Design a square-root procedure that uses this kind of end test. Does this work beer for small and large numbers? Exercise 1.8: Newton’s method for cube roots is based on the fact that if y is an approximation to the cube root of x, then a beer approximation is given by the value x/y 2 + 2y . 3 Use this formula to implement a cube-root procedure anal- ogous to the square-root procedure. (In Section 1.3.4 we will see how to implement Newton’s method in general as an abstraction of these square-root and cube-root procedures.) 1.1.8 Procedures as Black-Box Abstractions sqrt is our ﬁrst example of a process deﬁned by a set of mutually deﬁned procedures. Notice that the deﬁnition of sqrt-iter is recursive; that is, the procedure is deﬁned in terms of itself. e idea of being able to deﬁne a procedure in terms of itself may be disturbing; it may seem 33 sqrt | sqrt-iter / \ good-enough improve / \ \ square abs average Figure 1.2: Procedural decomposition of the sqrt program. unclear how such a “circular” deﬁnition could make sense at all, much less specify a well-deﬁned process to be carried out by a computer. is will be addressed more carefully in Section 1.2. But ﬁrst let’s consider some other important points illustrated by the sqrt example. Observe that the problem of computing square roots breaks up nat- urally into a number of subproblems: how to tell whether a guess is good enough, how to improve a guess, and so on. Each of these tasks is accomplished by a separate procedure. e entire sqrt program can be viewed as a cluster of procedures (shown in Figure 1.2) that mirrors the decomposition of the problem into subproblems. e importance of this decomposition strategy is not simply that one is dividing the program into parts. Aer all, we could take any large program and divide it into parts—the ﬁrst ten lines, the next ten lines, the next ten lines, and so on. Rather, it is crucial that each procedure ac- complishes an identiﬁable task that can be used as a module in deﬁning other procedures. For example, when we deﬁne the good-enough? pro- cedure in terms of square, we are able to regard the square procedure as a “black box.” We are not at that moment concerned with how the procedure computes its result, only with the fact that it computes the square. e details of how the square is computed can be suppressed, to be considered at a later time. Indeed, as far as the good-enough? pro- 34 cedure is concerned, square is not quite a procedure but rather an ab- straction of a procedure, a so-called procedural abstraction. At this level of abstraction, any procedure that computes the square is equally good. us, considering only the values they return, the following two procedures for squaring a number should be indistinguishable. Each takes a numerical argument and produces the square of that number as the value.25 (define (square x) (* x x)) (define (square x) (exp (double (log x)))) (define (double x) (+ x x)) So a procedure deﬁnition should be able to suppress detail. e users of the procedure may not have wrien the procedure themselves, but may have obtained it from another programmer as a black box. A user should not need to know how the procedure is implemented in order to use it. Local names One detail of a procedure’s implementation that should not maer to the user of the procedure is the implementer’s choice of names for the procedure’s formal parameters. us, the following procedures should not be distinguishable: (define (square x) (* x x)) (define (square y) (* y y)) 25 Itis not even clear which of these procedures is a more eﬃcient implementation. is depends upon the hardware available. ere are machines for which the “obvious” implementation is the less eﬃcient one. Consider a machine that has extensive tables of logarithms and antilogarithms stored in a very eﬃcient manner. 35 is principle—that the meaning of a procedure should be independent of the parameter names used by its author—seems on the surface to be self-evident, but its consequences are profound. e simplest conse- quence is that the parameter names of a procedure must be local to the body of the procedure. For example, we used square in the deﬁnition of good-enough? in our square-root procedure: (define (good-enough? guess x) (< (abs (- (square guess) x)) 0.001)) e intention of the author of good-enough? is to determine if the square of the ﬁrst argument is within a given tolerance of the second argument. We see that the author of good-enough? used the name guess to refer to the ﬁrst argument and x to refer to the second argument. e argument of square is guess. If the author of square used x (as above) to refer to that argument, we see that the x in good-enough? must be a diﬀerent x than the one in square. Running the procedure square must not aﬀect the value of x that is used by good-enough?, because that value of x may be needed by good-enough? aer square is done computing. If the parameters were not local to the bodies of their respective procedures, then the parameter x in square could be confused with the parameter x in good-enough?, and the behavior of good-enough? would depend upon which version of square we used. us, square would not be the black box we desired. A formal parameter of a procedure has a very special role in the procedure deﬁnition, in that it doesn’t maer what name the formal parameter has. Such a name is called a bound variable, and we say that the procedure deﬁnition binds its formal parameters. e meaning of a procedure deﬁnition is unchanged if a bound variable is consistently 36 renamed throughout the deﬁnition.26 If a variable is not bound, we say that it is free. e set of expressions for which a binding deﬁnes a name is called the scope of that name. In a procedure deﬁnition, the bound variables declared as the formal parameters of the procedure have the body of the procedure as their scope. In the deﬁnition of good-enough? above, guess and x are bound variables but <, -, abs, and square are free. e meaning of good- enough? should be independent of the names we choose for guess and x so long as they are distinct and diﬀerent from <, -, abs, and square. (If we renamed guess to abs we would have introduced a bug by cap- turing the variable abs. It would have changed from free to bound.) e meaning of good-enough? is not independent of the names of its free variables, however. It surely depends upon the fact (external to this def- inition) that the symbol abs names a procedure for computing the abso- lute value of a number. good-enough? will compute a diﬀerent function if we substitute cos for abs in its deﬁnition. Internal definitions and block structure We have one kind of name isolation available to us so far: e formal parameters of a procedure are local to the body of the procedure. e square-root program illustrates another way in which we would like to control the use of names. e existing program consists of separate procedures: (define (sqrt x) (sqrt-iter 1.0 x)) (define (sqrt-iter guess x) (if (good-enough? guess x) 26 e concept of consistent renaming is actually subtle and diﬃcult to deﬁne for- mally. Famous logicians have made embarrassing errors here. 37 guess (sqrt-iter (improve guess x) x))) (define (good-enough? guess x) (< (abs (- (square guess) x)) 0.001)) (define (improve guess x) (average guess (/ x guess))) e problem with this program is that the only procedure that is impor- tant to users of sqrt is sqrt. e other procedures (sqrt-iter, good- enough?, and improve) only cluer up their minds. ey may not deﬁne any other procedure called good-enough? as part of another program to work together with the square-root program, because sqrt needs it. e problem is especially severe in the construction of large systems by many separate programmers. For example, in the construction of a large library of numerical procedures, many numerical functions are computed as successive approximations and thus might have proce- dures named good-enough? and improve as auxiliary procedures. We would like to localize the subprocedures, hiding them inside sqrt so that sqrt could coexist with other successive approximations, each hav- ing its own private good-enough? procedure. To make this possible, we allow a procedure to have internal deﬁnitions that are local to that pro- cedure. For example, in the square-root problem we can write (define (sqrt x) (define (good-enough? guess x) (< (abs (- (square guess) x)) 0.001)) (define (improve guess x) (average guess (/ x guess))) (define (sqrt-iter guess x) (if (good-enough? guess x) guess (sqrt-iter (improve guess x) x))) (sqrt-iter 1.0 x)) 38 Such nesting of deﬁnitions, called block structure, is basically the right solution to the simplest name-packaging problem. But there is a bet- ter idea lurking here. In addition to internalizing the deﬁnitions of the auxiliary procedures, we can simplify them. Since x is bound in the deﬁ- nition of sqrt, the procedures good-enough?, improve, and sqrt-iter, which are deﬁned internally to sqrt, are in the scope of x. us, it is not necessary to pass x explicitly to each of these procedures. Instead, we allow x to be a free variable in the internal deﬁnitions, as shown be- low. en x gets its value from the argument with which the enclosing procedure sqrt is called. is discipline is called lexical scoping.27 (define (sqrt x) (define (good-enough? guess) (< (abs (- (square guess) x)) 0.001)) (define (improve guess) (average guess (/ x guess))) (define (sqrt-iter guess) (if (good-enough? guess) guess (sqrt-iter (improve guess)))) (sqrt-iter 1.0)) We will use block structure extensively to help us break up large pro- grams into tractable pieces.28 e idea of block structure originated with the programming language Algol 60. It appears in most advanced pro- gramming languages and is an important tool for helping to organize the construction of large programs. 27 Lexical scoping dictates that free variables in a procedure are taken to refer to bindings made by enclosing procedure deﬁnitions; that is, they are looked up in the environment in which the procedure was deﬁned. We will see how this works in detail in chapter 3 when we study environments and the detailed behavior of the interpreter. 28 Embedded deﬁnitions must come ﬁrst in a procedure body. e management is not responsible for the consequences of running programs that intertwine deﬁnition and use. 39 1.2 Procedures and the Processes They Generate We have now considered the elements of programming: We have used primitive arithmetic operations, we have combined these operations, and we have abstracted these composite operations by deﬁning them as compound procedures. But that is not enough to enable us to say that we know how to program. Our situation is analogous to that of someone who has learned the rules for how the pieces move in chess but knows nothing of typical openings, tactics, or strategy. Like the novice chess player, we don’t yet know the common paerns of usage in the do- main. We lack the knowledge of which moves are worth making (which procedures are worth deﬁning). We lack the experience to predict the consequences of making a move (executing a procedure). e ability to visualize the consequences of the actions under con- sideration is crucial to becoming an expert programmer, just as it is in any synthetic, creative activity. In becoming an expert photographer, for example, one must learn how to look at a scene and know how dark each region will appear on a print for each possible choice of exposure and development conditions. Only then can one reason backward, plan- ning framing, lighting, exposure, and development to obtain the desired eﬀects. So it is with programming, where we are planning the course of action to be taken by a process and where we control the process by means of a program. To become experts, we must learn to visualize the processes generated by various types of procedures. Only aer we have developed such a skill can we learn to reliably construct programs that exhibit the desired behavior. A procedure is a paern for the local evolution of a computational process. It speciﬁes how each stage of the process is built upon the previ- ous stage. We would like to be able to make statements about the overall, 40 or global, behavior of a process whose local evolution has been speciﬁed by a procedure. is is very diﬃcult to do in general, but we can at least try to describe some typical paerns of process evolution. In this section we will examine some common “shapes” for pro- cesses generated by simple procedures. We will also investigate the rates at which these processes consume the important computational resources of time and space. e procedures we will consider are very simple. eir role is like that played by test paerns in photography: as oversimpliﬁed prototypical paerns, rather than practical examples in their own right. 1.2.1 Linear Recursion and Iteration We begin by considering the factorial function, deﬁned by n! = n · (n − 1) · (n − 2) · · · 3 · 2 · 1. ere are many ways to compute factorials. One way is to make use of the observation that n! is equal to n times (n − 1)! for any positive integer n: n! = n · [(n − 1) · (n − 2) · · · 3 · 2 · 1] = n · (n − 1)!. us, we can compute n! by computing (n − 1)! and multiplying the result by n. If we add the stipulation that 1! is equal to 1, this observation translates directly into a procedure: (define (factorial n) (if (= n 1) 1 (* n (factorial (- n 1))))) 41 (factorial 6) (* 6 (factorial 5)) (* 6 (* 5 (factorial 4))) (* 6 (* 5 (* 4 (factorial 3)))) (* 6 (* 5 (* 4 (* 3 (factorial 2))))) (* 6 (* 5 (* 4 (* 3 (* 2 (factorial 1)))))) (* 6 (* 5 (* 4 (* 3 (* 2 1))))) (* 6 (* 5 (* 4 (* 3 2)))) (* 6 (* 5 (* 4 6))) (* 6 (* 5 24)) (* 6 120) 720 Figure 1.3: A linear recursive process for computing 6!. We can use the substitution model of Section 1.1.5 to watch this proce- dure in action computing 6!, as shown in Figure 1.3. Now let’s take a diﬀerent perspective on computing factorials. We could describe a rule for computing n! by specifying that we ﬁrst mul- tiply 1 by 2, then multiply the result by 3, then by 4, and so on until we reach n. More formally, we maintain a running product, together with a counter that counts from 1 up to n. We can describe the computation by saying that the counter and the product simultaneously change from one step to the next according to the rule product ← counter * product counter ← counter + 1 and stipulating that n! is the value of the product when the counter exceeds n. Once again, we can recast our description as a procedure for com- puting factorials:29 29 In a real program we would probably use the block structure introduced in the last section to hide the deﬁnition of fact-iter: 42 (factorial 6) (fact-iter 1 1 6) (fact-iter 1 2 6) (fact-iter 2 3 6) (fact-iter 6 4 6) (fact-iter 24 5 6) (fact-iter 120 6 6) (fact-iter 720 7 6) 720 Figure 1.4: A linear iterative process for computing 6!. (define (factorial n) (fact-iter 1 1 n)) (define (fact-iter product counter max-count) (if (> counter max-count) product (fact-iter (* counter product) (+ counter 1) max-count))) As before, we can use the substitution model to visualize the process of computing 6!, as shown in Figure 1.4. (define (factorial n) (define (iter product counter) (if (> counter n) product (iter (* counter product) (+ counter 1)))) (iter 1 1)) We avoided doing this here so as to minimize the number of things to think about at once. 43 Compare the two processes. From one point of view, they seem hardly diﬀerent at all. Both compute the same mathematical function on the same domain, and each requires a number of steps proportional to n to compute n!. Indeed, both processes even carry out the same sequence of multiplications, obtaining the same sequence of partial products. On the other hand, when we consider the “shapes” of the two processes, we ﬁnd that they evolve quite diﬀerently. Consider the ﬁrst process. e substitution model reveals a shape of expansion followed by contraction, indicated by the arrow in Figure 1.3. e expansion occurs as the process builds up a chain of deferred oper- ations (in this case, a chain of multiplications). e contraction occurs as the operations are actually performed. is type of process, charac- terized by a chain of deferred operations, is called a recursive process. Carrying out this process requires that the interpreter keep track of the operations to be performed later on. In the computation of n!, the length of the chain of deferred multiplications, and hence the amount of infor- mation needed to keep track of it, grows linearly with n (is proportional to n), just like the number of steps. Such a process is called a linear re- cursive process. By contrast, the second process does not grow and shrink. At each step, all we need to keep track of, for any n, are the current values of the variables product, counter, and max-count. We call this an iterative process. In general, an iterative process is one whose state can be sum- marized by a ﬁxed number of state variables, together with a ﬁxed rule that describes how the state variables should be updated as the process moves from state to state and an (optional) end test that speciﬁes con- ditions under which the process should terminate. In computing n!, the number of steps required grows linearly with n. Such a process is called a linear iterative process. 44 e contrast between the two processes can be seen in another way. In the iterative case, the program variables provide a complete descrip- tion of the state of the process at any point. If we stopped the compu- tation between steps, all we would need to do to resume the computa- tion is to supply the interpreter with the values of the three program variables. Not so with the recursive process. In this case there is some additional “hidden” information, maintained by the interpreter and not contained in the program variables, which indicates “where the process is” in negotiating the chain of deferred operations. e longer the chain, the more information must be maintained.30 In contrasting iteration and recursion, we must be careful not to confuse the notion of a recursive process with the notion of a recursive procedure. When we describe a procedure as recursive, we are referring to the syntactic fact that the procedure deﬁnition refers (either directly or indirectly) to the procedure itself. But when we describe a process as following a paern that is, say, linearly recursive, we are speaking about how the process evolves, not about the syntax of how a procedure is wrien. It may seem disturbing that we refer to a recursive procedure such as fact-iter as generating an iterative process. However, the pro- cess really is iterative: Its state is captured completely by its three state variables, and an interpreter need keep track of only three variables in order to execute the process. One reason that the distinction between process and procedure may be confusing is that most implementations of common languages (in- cluding Ada, Pascal, and C) are designed in such a way that the interpre- tation of any recursive procedure consumes an amount of memory that 30 When we discuss the implementation of procedures on register machines in Chap- ter 5, we will see that any iterative process can be realized “in hardware” as a machine that has a ﬁxed set of registers and no auxiliary memory. In contrast, realizing a re- cursive process requires a machine that uses an auxiliary data structure known as a stack. 45 grows with the number of procedure calls, even when the process de- scribed is, in principle, iterative. As a consequence, these languages can describe iterative processes only by resorting to special-purpose “loop- ing constructs” such as do, repeat, until, for, and while. e imple- mentation of Scheme we shall consider in Chapter 5 does not share this defect. It will execute an iterative process in constant space, even if the iterative process is described by a recursive procedure. An implemen- tation with this property is called tail-recursive. With a tail-recursive implementation, iteration can be expressed using the ordinary proce- dure call mechanism, so that special iteration constructs are useful only as syntactic sugar.31 Exercise 1.9: Each of the following two procedures deﬁnes a method for adding two positive integers in terms of the procedures inc, which increments its argument by 1, and dec, which decrements its argument by 1. (define (+ a b) (if (= a 0) b (inc (+ (dec a) b)))) (define (+ a b) (if (= a 0) b (+ (dec a) (inc b)))) Using the substitution model, illustrate the process gener- ated by each procedure in evaluating (+ 4 5). Are these processes iterative or recursive? 31 Tail recursion has long been known as a compiler optimization trick. A coherent semantic basis for tail recursion was provided by Carl Hewi (1977), who explained it in terms of the “message-passing” model of computation that we shall discuss in Chapter 3. Inspired by this, Gerald Jay Sussman and Guy Lewis Steele Jr. (see Steele and Sussman 1975) constructed a tail-recursive interpreter for Scheme. Steele later showed how tail recursion is a consequence of the natural way to compile procedure calls (Steele 1977). e standard for Scheme requires that Scheme implementations be tail-recursive. 46 Exercise 1.10: e following procedure computes a math- ematical function called Ackermann’s function. (define (A x y) (cond ((= y 0) 0) ((= x 0) (* 2 y)) ((= y 1) 2) (else (A (- x 1) (A x (- y 1)))))) What are the values of the following expressions? (A 1 10) (A 2 4) (A 3 3) Consider the following procedures, where A is the proce- dure deﬁned above: (define (f n) (A 0 n)) (define (g n) (A 1 n)) (define (h n) (A 2 n)) (define (k n) (* 5 n n)) Give concise mathematical deﬁnitions for the functions com- puted by the procedures f, g, and h for positive integer val- ues of n. For example, (k n) computes 5n 2 . 1.2.2 Tree Recursion Another common paern of computation is called tree recursion. As an example, consider computing the sequence of Fibonacci numbers, in which each number is the sum of the preceding two: 0, 1, 1, 2, 3, 5, 8, 13, 21, . . . . 47 In general, the Fibonacci numbers can be deﬁned by the rule 0 if n = 0, Fib(n) = 1 if n = 1, Fib(n − 1) + Fib(n − 2) otherwise. We can immediately translate this deﬁnition into a recursive procedure for computing Fibonacci numbers: (define (fib n) (cond ((= n 0) 0) ((= n 1) 1) (else (+ (fib (- n 1)) (fib (- n 2)))))) Consider the paern of this computation. To compute (fib 5), we com- pute (fib 4) and (fib 3). To compute (fib 4), we compute (fib 3) and (fib 2). In general, the evolved process looks like a tree, as shown in Figure 1.5. Notice that the branches split into two at each level (ex- cept at the boom); this reﬂects the fact that the fib procedure calls itself twice each time it is invoked. is procedure is instructive as a prototypical tree recursion, but it is a terrible way to compute Fibonacci numbers because it does so much redundant computation. Notice in Figure 1.5 that the entire computation of (fib 3)—almost half the work—is duplicated. In fact, it is not hard to show that the number of times the procedure will compute (fib 1) or (fib 0) (the number of leaves in the above tree, in general) is precisely Fib(n + 1). To get an idea of how bad this is, one can show that the value of Fib(n) grows exponentially with√n. More precisely (see Exercise 1.13), Fib(n) is the closest integer to ϕ n / 5, where √ 1+ 5 ϕ= ≈ 1.6180 2 48 fib 5 fib 4 fib 3 fib 3 fib 2 fib 2 fib 1 1 fib 2 fib 1 fib 1 fib 0 fib 1 fib 0 1 1 0 1 0 fib 1 fib 0 1 0 Figure 1.5: e tree-recursive process generated in com- puting (fib 5). is the golden ratio, which satisﬁes the equation ϕ 2 = ϕ + 1. us, the process uses a number of steps that grows exponentially with the input. On the other hand, the space required grows only linearly with the input, because we need keep track only of which nodes are above us in the tree at any point in the computation. In general, the number of steps required by a tree-recursive process will be propor- tional to the number of nodes in the tree, while the space required will be proportional to the maximum depth of the tree. We can also formulate an iterative process for computing the Fi- bonacci numbers. e idea is to use a pair of integers a and b, initialized to Fib(1) = 1 and Fib(0) = 0, and to repeatedly apply the simultaneous 49 transformations a ← a + b, b ← a. It is not hard to show that, aer applying this transformation n times, a and b will be equal, respectively, to Fib(n + 1) and Fib(n). us, we can compute Fibonacci numbers iteratively using the procedure (define (fib n) (fib-iter 1 0 n)) (define (fib-iter a b count) (if (= count 0) b (fib-iter (+ a b) a (- count 1)))) is second method for computing Fib(n) is a linear iteration. e diﬀer- ence in number of steps required by the two methods—one linear in n, one growing as fast as Fib(n) itself—is enormous, even for small inputs. One should not conclude from this that tree-recursive processes are useless. When we consider processes that operate on hierarchically structured data rather than numbers, we will ﬁnd that tree recursion is a natural and powerful tool.32 But even in numerical operations, tree- recursive processes can be useful in helping us to understand and de- sign programs. For instance, although the ﬁrst fib procedure is much less eﬃcient than the second one, it is more straightforward, being lile more than a translation into Lisp of the deﬁnition of the Fibonacci se- quence. To formulate the iterative algorithm required noticing that the computation could be recast as an iteration with three state variables. 32 An example of this was hinted at in Section 1.1.3. e interpreter itself evaluates expressions using a tree-recursive process. 50 Example: Counting change It takes only a bit of cleverness to come up with the iterative Fibonacci algorithm. In contrast, consider the following problem: How many dif- ferent ways can we make change of $1.00, given half-dollars, quarters, dimes, nickels, and pennies? More generally, can we write a procedure to compute the number of ways to change any given amount of money? is problem has a simple solution as a recursive procedure. Sup- pose we think of the types of coins available as arranged in some order. en the following relation holds: e number of ways to change amount a using n kinds of coins equals • the number of ways to change amount a using all but the ﬁrst kind of coin, plus • the number of ways to change amount a − d using all n kinds of coins, where d is the denomination of the ﬁrst kind of coin. To see why this is true, observe that the ways to make change can be divided into two groups: those that do not use any of the ﬁrst kind of coin, and those that do. erefore, the total number of ways to make change for some amount is equal to the number of ways to make change for the amount without using any of the ﬁrst kind of coin, plus the number of ways to make change assuming that we do use the ﬁrst kind of coin. But the laer number is equal to the number of ways to make change for the amount that remains aer using a coin of the ﬁrst kind. us, we can recursively reduce the problem of changing a given amount to the problem of changing smaller amounts using fewer kinds of coins. Consider this reduction rule carefully, and convince yourself 51 that we can use it to describe an algorithm if we specify the following degenerate cases:33 • If a is exactly 0, we should count that as 1 way to make change. • If a is less than 0, we should count that as 0 ways to make change. • If n is 0, we should count that as 0 ways to make change. We can easily translate this description into a recursive procedure: (define (count-change amount) (cc amount 5)) (define (cc amount kinds-of-coins) (cond ((= amount 0) 1) ((or (< amount 0) (= kinds-of-coins 0)) 0) (else (+ (cc amount (- kinds-of-coins 1)) (cc (- amount (first-denomination kinds-of-coins)) kinds-of-coins))))) (define (first-denomination kinds-of-coins) (cond ((= kinds-of-coins 1) 1) ((= kinds-of-coins 2) 5) ((= kinds-of-coins 3) 10) ((= kinds-of-coins 4) 25) ((= kinds-of-coins 5) 50))) (e first-denomination procedure takes as input the number of kinds of coins available and returns the denomination of the ﬁrst kind. Here we are thinking of the coins as arranged in order from largest to small- est, but any order would do as well.) We can now answer our original question about changing a dollar: 33 For example, work through in detail how the reduction rule applies to the problem of making change for 10 cents using pennies and nickels. 52 (count-change 100) 292 count-change generates a tree-recursive process with redundancies sim- ilar to those in our ﬁrst implementation of fib. (It will take quite a while for that 292 to be computed.) On the other hand, it is not obvious how to design a beer algorithm for computing the result, and we leave this problem as a challenge. e observation that a tree-recursive process may be highly ineﬃcient but oen easy to specify and understand has led people to propose that one could get the best of both worlds by designing a “smart compiler” that could transform tree-recursive pro- cedures into more eﬃcient procedures that compute the same result.34 Exercise 1.11: A function f is deﬁned by the rule that n if n < 3, f (n) = f (n − 1) + 2f (n − 2) + 3f (n − 3) if n ≥ 3. Write a procedure that computes f by means of a recursive process. Write a procedure that computes f by means of an iterative process. Exercise 1.12: e following paern of numbers is called Pascal’s triangle. 34 One approach to coping with redundant computations is to arrange maers so that we automatically construct a table of values as they are computed. Each time we are asked to apply the procedure to some argument, we ﬁrst look to see if the value is already stored in the table, in which case we avoid performing the redundant com- putation. is strategy, known as tabulation or memoization, can be implemented in a straightforward way. Tabulation can sometimes be used to transform processes that require an exponential number of steps (such as count-change) into processes whose space and time requirements grow linearly with the input. See Exercise 3.27. 53 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 . . . e numbers at the edge of the triangle are all 1, and each number inside the triangle is the sum of the two numbers above it.35 Write a procedure that computes elements of Pascal’s triangle by means of a recursive process. Exercise √ 1.13: Prove that √ Fib(n) is the closest integer√ to ϕ / 5, where ϕ = (1 + 5)/2. Hint: Let ψ = (1 − 5)/2. n Use induction and the deﬁnition of the Fibonacci numbers√ (see Section 1.2.2) to prove that Fib(n) = (ϕ n − ψ n )/ 5. 1.2.3 Orders of Growth e previous examples illustrate that processes can diﬀer considerably in the rates at which they consume computational resources. One con- venient way to describe this diﬀerence is to use the notion of order of growth to obtain a gross measure of the resources required by a process as the inputs become larger. 35 e elements of Pascal’s triangle are called the binomial coeﬃcients, because the n throw consists of the coeﬃcients of the terms in the expansion of (x + y)n . is pat- tern for computing the coeﬃcients appeared in Blaise Pascal’s 1653 seminal work on probability theory, Traité du triangle arithmétique. According to Knuth (1973), the same paern appears in the Szu-yuen Yü-chien (“e Precious Mirror of the Four Elements”), published by the Chinese mathematician Chu Shih-chieh in 1303, in the works of the twelh-century Persian poet and mathematician Omar Khayyam, and in the works of the twelh-century Hindu mathematician Bháscara Áchárya. 54 Let n be a parameter that measures the size of the problem, and let R(n) be the amount of resources the process requires for a problem of size n. In our previous examples we took n to be the number for which a given function is to be computed, but there are other possibilities. For instance, if our goal is to compute an approximation to the square root of a number, we might take n to be the number of digits accuracy required. For matrix multiplication we might take n to be the number of rows in the matrices. In general there are a number of properties of the problem with respect to which it will be desirable to analyze a given process. Similarly, R(n) might measure the number of internal storage registers used, the number of elementary machine operations performed, and so on. In computers that do only a ﬁxed number of operations at a time, the time required will be proportional to the number of elementary machine operations performed. We say that R(n) has order of growth Θ(f (n)), wrien R(n) = Θ(f (n)) (pronounced “theta of f (n)”), if there are positive constants k 1 and k 2 independent of n such that k 1 f (n) ≤ R(n) ≤ k 2 f (n) for any suﬃciently large value of n. (In other words, for large n, the value R(n) is sandwiched between k 1 f (n) and k 2 f (n).) For instance, with the linear recursive process for computing facto- rial described in Section 1.2.1 the number of steps grows proportionally to the input n. us, the steps required for this process grows as Θ(n). We also saw that the space required grows as Θ(n). For the iterative factorial, the number of steps is still Θ(n) but the space is Θ(1)—that is, constant.36 e tree-recursive Fibonacci computation requires Θ(ϕ n ) 36 ese statements mask a great deal of oversimpliﬁcation. For instance, if we count process steps as “machine operations” we are making the assumption that the number of machine operations needed to perform, say, a multiplication is independent of the size of the numbers to be multiplied, which is false if the numbers are suﬃciently large. Similar remarks hold for the estimates of space. Like the design and description of a process, the analysis of a process can be carried out at various levels of abstraction. 55 steps and space Θ(n), where ϕ is the golden ratio described in Section 1.2.2. Orders of growth provide only a crude description of the behavior of a process. For example, a process requiring n 2 steps and a process requiring 1000n 2 steps and a process requiring 3n 2 + 10n + 17 steps all have Θ(n 2 ) order of growth. On the other hand, order of growth provides a useful indication of how we may expect the behavior of the process to change as we change the size of the problem. For a Θ(n) (linear) process, doubling the size will roughly double the amount of resources used. For an exponential process, each increment in problem size will multiply the resource utilization by a constant factor. In the remainder of Section 1.2 we will examine two algorithms whose order of growth is logarithmic, so that doubling the problem size increases the resource requirement by a constant amount. Exercise 1.14: Draw the tree illustrating the process gen- erated by the count-change procedure of Section 1.2.2 in making change for 11 cents. What are the orders of growth of the space and number of steps used by this process as the amount to be changed increases? Exercise 1.15: e sine of an angle (speciﬁed in radians) can be computed by making use of the approximation sin x ≈ x if x is suﬃciently small, and the trigonometric identity x x sin x = 3 sin − 4 sin3 3 3 to reduce the size of the argument of sin. (For purposes of this exercise an angle is considered “suﬃciently small” if its magnitude is not greater than 0.1 radians.) ese ideas are incorporated in the following procedures: 56 (define (cube x) (* x x x)) (define (p x) (- (* 3 x) (* 4 (cube x)))) (define (sine angle) (if (not (> (abs angle) 0.1)) angle (p (sine (/ angle 3.0))))) a. How many times is the procedure p applied when (sine 12.15) is evaluated? b. What is the order of growth in space and number of steps (as a function of a) used by the process generated by the sine procedure when (sine a) is evaluated? 1.2.4 Exponentiation Consider the problem of computing the exponential of a given number. We would like a procedure that takes as arguments a base b and a posi- tive integer exponent n and computes b n . One way to do this is via the recursive deﬁnition b n = b · b n −1 , b 0 = 1, which translates readily into the procedure (define (expt b n) (if (= n 0) 1 (* b (expt b (- n 1))))) is is a linear recursive process, which requires Θ(n) steps and Θ(n) space. Just as with factorial, we can readily formulate an equivalent lin- ear iteration: 57 (define (expt b n) (expt-iter b n 1)) (define (expt-iter b counter product) (if (= counter 0) product (expt-iter b (- counter 1) (* b product)))) is version requires Θ(n) steps and Θ(1) space. We can compute exponentials in fewer steps by using successive squaring. For instance, rather than computing b 8 as b · (b · (b · (b · (b · (b · (b · b)))))) , we can compute it using three multiplications: b 2 = b · b, b4 = b2 · b2 , b8 = b4 · b4. is method works ﬁne for exponents that are powers of 2. We can also take advantage of successive squaring in computing exponentials in general if we use the rule b n = (b n/2 )2 if n is even, b n = b · b n −1 if n is odd. We can express this method as a procedure: (define (fast-expt b n) (cond ((= n 0) 1) ((even? n) (square (fast-expt b (/ n 2)))) (else (* b (fast-expt b (- n 1)))))) 58 where the predicate to test whether an integer is even is deﬁned in terms of the primitive procedure remainder by (define (even? n) (= (remainder n 2) 0)) e process evolved by fast-expt grows logarithmically with n in both space and number of steps. To see this, observe that computing b 2n us- ing fast-expt requires only one more multiplication than computing b n . e size of the exponent we can compute therefore doubles (approx- imately) with every new multiplication we are allowed. us, the num- ber of multiplications required for an exponent of n grows about as fast as the logarithm of n to the base 2. e process has Θ(log n) growth.37 e diﬀerence between Θ(log n) growth and Θ(n) growth becomes striking as n becomes large. For example, fast-expt for n = 1000 re- quires only 14 multiplications.38 It is also possible to use the idea of successive squaring to devise an iterative algorithm that computes ex- ponentials with a logarithmic number of steps (see Exercise 1.16), al- though, as is oen the case with iterative algorithms, this is not wrien down so straightforwardly as the recursive algorithm.39 Exercise 1.16: Design a procedure that evolves an itera- tive exponentiation process that uses successive squaring 37 More precisely, the number of multiplications required is equal to 1 less than the log base 2 of n plus the number of ones in the binary representation of n. is total is always less than twice the log base 2 of n. e arbitrary constants k 1 and k 2 in the deﬁnition of order notation imply that, for a logarithmic process, the base to which logarithms are taken does not maer, so all such processes are described as Θ(log n). 38 You may wonder why anyone would care about raising numbers to the 1000th power. See Section 1.2.6. 39 is iterative algorithm is ancient. It appears in the Chandah-sutra by Áchárya Pingala, wrien before 200 .. See Knuth 1981, section 4.6.3, for a full discussion and analysis of this and other methods of exponentiation. 59 and uses a logarithmic number of steps, as does fast-expt. (Hint: Using the observation that (b n/2 )2 = (b 2 )n/2 , keep, along with the exponent n and the base b, an additional state variable a, and deﬁne the state transformation in such a way that the product ab n is unchanged from state to state. At the beginning of the process a is taken to be 1, and the answer is given by the value of a at the end of the process. In general, the technique of deﬁning an invariant quantity that remains unchanged from state to state is a powerful way to think about the design of iterative algorithms.) Exercise 1.17: e exponentiation algorithms in this sec- tion are based on performing exponentiation by means of repeated multiplication. In a similar way, one can perform integer multiplication by means of repeated addition. e following multiplication procedure (in which it is assumed that our language can only add, not multiply) is analogous to the expt procedure: (define (* a b) (if (= b 0) 0 (+ a (* a (- b 1))))) is algorithm takes a number of steps that is linear in b. Now suppose we include, together with addition, opera- tions double, which doubles an integer, and halve, which divides an (even) integer by 2. Using these, design a mul- tiplication procedure analogous to fast-expt that uses a logarithmic number of steps. 60 Exercise 1.18: Using the results of Exercise 1.16 and Exer- cise 1.17, devise a procedure that generates an iterative pro- cess for multiplying two integers in terms of adding, dou- bling, and halving and uses a logarithmic number of steps.40 Exercise 1.19: ere is a clever algorithm for computing the Fibonacci numbers in a logarithmic number of steps. Recall the transformation of the state variables a and b in the fib-iter process of Section 1.2.2: a ← a +b and b ← a. Call this transformation T , and observe that applying T over and over again n times, starting with 1 and 0, produces the pair Fib(n + 1) and Fib(n). In other words, the Fibonacci numbers are produced by applying T n , the n th power of the transformationT , starting with the pair (1, 0). Now consider T to be the special case of p = 0 and q = 1 in a family of transformations Tpq , where Tpq transforms the pair (a, b) according to a ← bq + aq + ap and b ← bp + aq. Show that if we apply such a transformation Tpq twice, the eﬀect is the same as using a single transformation Tp ′q ′ of the same form, and compute p ′ and q ′ in terms of p and q. is gives us an explicit way to square these transformations, and thus we can compute T n using successive squaring, as in the fast-expt procedure. Put this all together to com- plete the following procedure, which runs in a logarithmic number of steps:41 40 is algorithm, which is sometimes known as the “Russian peasant method” of multiplication, is ancient. Examples of its use are found in the Rhind Papyrus, one of the two oldest mathematical documents in existence, wrien about 1700 .. (and copied from an even older document) by an Egyptian scribe named A’h-mose. 41 is exercise was suggested to us by Joe Stoy, based on an example in Kaldewaij 1990. 61 (define (fib n) (fib-iter 1 0 0 1 n)) (define (fib-iter a b p q count) (cond ((= count 0) b) ((even? count) (fib-iter a b ⟨??⟩ ; compute p ′ ⟨??⟩ ; compute q ′ (/ count 2))) (else (fib-iter (+ (* b q) (* a q) (* a p)) (+ (* b p) (* a q)) p q (- count 1))))) 1.2.5 Greatest Common Divisors e greatest common divisor () of two integers a and b is deﬁned to be the largest integer that divides both a and b with no remainder. For example, the of 16 and 28 is 4. In Chapter 2, when we investigate how to implement rational-number arithmetic, we will need to be able to compute s in order to reduce rational numbers to lowest terms. (To reduce a rational number to lowest terms, we must divide both the numerator and the denominator by their . For example, 16/28 re- duces to 4/7.) One way to ﬁnd the of two integers is to factor them and search for common factors, but there is a famous algorithm that is much more eﬃcient. e idea of the algorithm is based on the observation that, if r is the remainder when a is divided by b, then the common divisors of a and b are precisely the same as the common divisors of b and r . us, we can 62 use the equation GCD(a,b) = GCD(b,r) to successively reduce the problem of computing a to the problem of computing the of smaller and smaller pairs of integers. For ex- ample, GCD(206,40) = GCD(40,6) = GCD(6,4) = GCD(4,2) = GCD(2,0) = 2 reduces (, ) to (, ), which is 2. It is possible to show that starting with any two positive integers and performing repeated reduc- tions will always eventually produce a pair where the second number is 0. en the is the other number in the pair. is method for com- puting the is known as Euclid’s Algorithm.42 It is easy to express Euclid’s Algorithm as a procedure: (define (gcd a b) (if (= b 0) a (gcd b (remainder a b)))) is generates an iterative process, whose number of steps grows as the logarithm of the numbers involved. 42 Euclid’sAlgorithm is so called because it appears in Euclid’s Elements (Book 7, ca. 300 ..). According to Knuth (1973), it can be considered the oldest known nontrivial algorithm. e ancient Egyptian method of multiplication (Exercise 1.18) is surely older, but, as Knuth explains, Euclid’s algorithm is the oldest known to have been presented as a general algorithm, rather than as a set of illustrative examples. 63 e fact that the number of steps required by Euclid’s Algorithm has logarithmic growth bears an interesting relation to the Fibonacci numbers: Lamé’s Theorem: If Euclid’s Algorithm requires k steps to compute the of some pair, then the smaller number in the pair must be greater than or equal to the k th Fibonacci number.43 We can use this theorem to get an order-of-growth estimate for Euclid’s Algorithm. Let n be the smaller of the two inputs to the procedure. √ If the process takes k steps, then we must have n ≥ Fib(k) ≈ ϕ k / 5. erefore the number of steps k grows as the logarithm (to the base ϕ) of n. Hence, the order of growth is Θ(log n). 43 is theorem was proved in 1845 by Gabriel Lamé, a French mathematician and engineer known chieﬂy for his contributions to mathematical physics. To prove the theorem, we consider pairs (a k , b k ), where a k ≥ b k , for which Euclid’s Algorithm terminates in k steps. e proof is based on the claim that, if (a k +1 , b k +1 ) → (a k , b k ) → (a k −1 , bk −1 ) are three successive pairs in the reduction process, then we must have bk +1 ≥ bk + b k −1 . To verify the claim, consider that a reduction step is deﬁned by applying the transformation a k −1 = bk , bk −1 = remainder of a k divided by b k . e second equation means that a k = qbk + bk −1 for some positive integer q. And since q must be at least 1 we have a k = qbk + b k −1 ≥ bk + bk −1 . But in the previous reduction step we have b k +1 = a k . erefore, bk +1 = a k ≥ b k +bk −1 . is veriﬁes the claim. Now we can prove the theorem by induction on k, the number of steps that the algorithm requires to terminate. e result is true for k = 1, since this merely requires that b be at least as large as Fib(1) = 1. Now, assume that the result is true for all integers less than or equal to k and establish the result for k + 1. Let (a k +1 , bk +1 ) → (a k , bk ) → (a k −1 , bk −1 ) be successive pairs in the reduction process. By our induction hypotheses, we have bk −1 ≥ Fib(k − 1) and bk ≥ Fib(k). us, applying the claim we just proved together with the deﬁnition of the Fibonacci numbers gives bk +1 ≥ bk + b k −1 ≥ Fib(k) + Fib(k − 1) = Fib(k + 1), which completes the proof of Lamé’s eorem. 64 Exercise 1.20: e process that a procedure generates is of course dependent on the rules used by the interpreter. As an example, consider the iterative gcd procedure given above. Suppose we were to interpret this procedure using normal-order evaluation, as discussed in Section 1.1.5. (e normal-order-evaluation rule for if is described in Exercise 1.5.) Using the substitution method (for normal order), illus- trate the process generated in evaluating (gcd 206 40) and indicate the remainder operations that are actually per- formed. How many remainder operations are actually per- formed in the normal-order evaluation of (gcd 206 40)? In the applicative-order evaluation? 1.2.6 Example: Testing for Primality is section describes two methods for checking the primality of an in- √ teger n, one with order of growth Θ( n), and a “probabilistic” algorithm with order of growth Θ(log n). e exercises at the end of this section suggest programming projects based on these algorithms. Searching for divisors Since ancient times, mathematicians have been fascinated by problems concerning prime numbers, and many people have worked on the prob- lem of determining ways to test if numbers are prime. One way to test if a number is prime is to ﬁnd the number’s divisors. e following pro- gram ﬁnds the smallest integral divisor (greater than 1) of a given num- ber n. It does this in a straightforward way, by testing n for divisibility by successive integers starting with 2. (define (smallest-divisor n) (find-divisor n 2)) 65 (define (find-divisor n test-divisor) (cond ((> (square test-divisor) n) n) ((divides? test-divisor n) test-divisor) (else (find-divisor n (+ test-divisor 1))))) (define (divides? a b) (= (remainder b a) 0)) We can test whether a number is prime as follows: n is prime if and only if n is its own smallest divisor. (define (prime? n) (= n (smallest-divisor n))) e end test for find-divisor is based on the fact that if n is not prime it √ must have a divisor less than or equal to n.44 is means that the algo- √ rithm need only test divisors between 1 and n. Consequently, the num- ber of steps required to identify n as prime will have order of growth √ Θ( n). The Fermat test e Θ(log n) primality test is based on a result from number theory known as Fermat’s Lile eorem.45 44 If d √ is a divisor of n, then so is n/d. But d and n/d cannot both be greater than n. 45 Pierre de Fermat (1601-1665) is considered to be the founder of modern number the- ory. He obtained many important number-theoretic results, but he usually announced just the results, without providing his proofs. Fermat’s Lile eorem was stated in a leer he wrote in 1640. e ﬁrst published proof was given by Euler in 1736 (and an earlier, identical proof was discovered in the unpublished manuscripts of Leibniz). e most famous of Fermat’s results—known as Fermat’s Last eorem—was joed down in 1637 in his copy of the book Arithmetic (by the third-century Greek mathematician Diophantus) with the remark “I have discovered a truly remarkable proof, but this mar- gin is too small to contain it.” Finding a proof of Fermat’s Last eorem became one of the most famous challenges in number theory. A complete solution was ﬁnally given in 1995 by Andrew Wiles of Princeton University. 66 Fermat’s Lile Theorem: If n is a prime number and a is any positive integer less than n, then a raised to the n th power is congruent to a modulo n. (Two numbers are said to be congruent modulo n if they both have the same remainder when divided by n. e remainder of a number a when divided by n is also referred to as the remainder of a modulo n, or simply as a modulo n.) If n is not prime, then, in general, most of the numbers a < n will not satisfy the above relation. is leads to the following algorithm for testing primality: Given a number n, pick a random number a < n and compute the remainder of an modulo n. If the result is not equal to a, then n is certainly not prime. If it is a, then chances are good that n is prime. Now pick another random number a and test it with the same method. If it also satisﬁes the equation, then we can be even more con- ﬁdent that n is prime. By trying more and more values of a, we can increase our conﬁdence in the result. is algorithm is known as the Fermat test. To implement the Fermat test, we need a procedure that computes the exponential of a number modulo another number: (define (expmod base exp m) (cond ((= exp 0) 1) ((even? exp) (remainder (square (expmod base (/ exp 2) m)) m)) (else (remainder (* base (expmod base (- exp 1) m)) m)))) 67 is is very similar to the fast-expt procedure of Section 1.2.4. It uses successive squaring, so that the number of steps grows logarithmically with the exponent.46 e Fermat test is performed by choosing at random a number a be- tween 1 and n −1 inclusive and checking whether the remainder modulo n of the n th power of a is equal to a. e random number a is chosen us- ing the procedure random, which we assume is included as a primitive in Scheme. random returns a nonnegative integer less than its integer input. Hence, to obtain a random number between 1 and n − 1, we call random with an input of n − 1 and add 1 to the result: (define (fermat-test n) (define (try-it a) (= (expmod a n n) a)) (try-it (+ 1 (random (- n 1))))) e following procedure runs the test a given number of times, as spec- iﬁed by a parameter. Its value is true if the test succeeds every time, and false otherwise. (define (fast-prime? n times) (cond ((= times 0) true) ((fermat-test n) (fast-prime? n (- times 1))) (else false))) 46 e reduction steps in the cases where the exponent e is greater than 1 are based on the fact that, for any integers x , y, and m, we can ﬁnd the remainder of x times y modulo m by computing separately the remainders of x modulo m and y modulo m, multiplying these, and then taking the remainder of the result modulo m. For instance, in the case where e is even, we compute the remainder of b e/2 modulo m, square this, and take the remainder modulo m. is technique is useful because it means we can perform our computation without ever having to deal with numbers much larger than m. (Compare Exercise 1.25.) 68 Probabilistic methods e Fermat test diﬀers in character from most familiar algorithms, in which one computes an answer that is guaranteed to be correct. Here, the answer obtained is only probably correct. More precisely, if n ever fails the Fermat test, we can be certain that n is not prime. But the fact that n passes the test, while an extremely strong indication, is still not a guarantee that n is prime. What we would like to say is that for any number n, if we perform the test enough times and ﬁnd that n always passes the test, then the probability of error in our primality test can be made as small as we like. Unfortunately, this assertion is not quite correct. ere do exist num- bers that fool the Fermat test: numbers n that are not prime and yet have the property that an is congruent to a modulo n for all integers a < n. Such numbers are extremely rare, so the Fermat test is quite reliable in practice.47 ere are variations of the Fermat test that cannot be fooled. In these tests, as with the Fermat method, one tests the primality of an integer n by choosing a random integer a < n and checking some condition that depends upon n and a. (See Exercise 1.28 for an example of such a test.) On the other hand, in contrast to the Fermat test, one can prove that, for any n, the condition does not hold for most of the integers a < n unless n is prime. us, if n passes the test for some random choice of 47 Numbers that fool the Fermat test are called Carmichael numbers, and lile is known about them other than that they are extremely rare. ere are 255 Carmichael numbers below 100,000,000. e smallest few are 561, 1105, 1729, 2465, 2821, and 6601. In testing primality of very large numbers chosen at random, the chance of stumbling upon a value that fools the Fermat test is less than the chance that cosmic radiation will cause the computer to make an error in carrying out a “correct” algorithm. Considering an algorithm to be inadequate for the ﬁrst reason but not for the second illustrates the diﬀerence between mathematics and engineering. 69 a, the chances are beer than even that n is prime. If n passes the test for two random choices of a, the chances are beer than 3 out of 4 that n is prime. By running the test with more and more randomly chosen values of a we can make the probability of error as small as we like. e existence of tests for which one can prove that the chance of error becomes arbitrarily small has sparked interest in algorithms of this type, which have come to be known as probabilistic algorithms. ere is a great deal of research activity in this area, and probabilistic algorithms have been fruitfully applied to many ﬁelds.48 Exercise 1.21: Use the smallest-divisor procedure to ﬁnd the smallest divisor of each of the following numbers: 199, 1999, 19999. Exercise 1.22: Most Lisp implementations include a prim- itive called runtime that returns an integer that speciﬁes the amount of time the system has been running (mea- sured, for example, in microseconds). e following timed- prime-test procedure, when called with an integer n, prints n and checks to see if n is prime. If n is prime, the procedure prints three asterisks followed by the amount of time used in performing the test. 48 One of the most striking applications of probabilistic prime testing has been to the ﬁeld of cryptography. Although it is now computationally infeasible to factor an arbi- trary 200-digit number, the primality of such a number can be checked in a few seconds with the Fermat test. is fact forms the basis of a technique for constructing “unbreak- able codes” suggested by Rivest et al. (1977). e resulting RSA algorithm has become a widely used technique for enhancing the security of electronic communications. Be- cause of this and related developments, the study of prime numbers, once considered the epitome of a topic in “pure” mathematics to be studied only for its own sake, now turns out to have important practical applications to cryptography, electronic funds transfer, and information retrieval. 70 (define (timed-prime-test n) (newline) (display n) (start-prime-test n (runtime))) (define (start-prime-test n start-time) (if (prime? n) (report-prime (- (runtime) start-time)))) (define (report-prime elapsed-time) (display " *** ") (display elapsed-time)) Using this procedure, write a procedure search-for-primes that checks the primality of consecutive odd integers in a speciﬁed range. Use your procedure to ﬁnd the three small- est primes larger than 1000; larger than 10,000; larger than 100,000; larger than 1,000,000. Note the time needed to test each prime. Since the testing algorithm has order of growth √ of Θ( n), you should expect√ that testing for primes around 10,000 should take about 10 times as long as testing for primes around 1000. Do your timing data bear this out? How well do the data for 100,000 and 1,000,000 support the √ Θ( n) prediction? Is your result compatible with the notion that programs on your machine run in time proportional to the number of steps required for the computation? Exercise 1.23: e smallest-divisor procedure shown at the start of this section does lots of needless testing: Aer it checks to see if the number is divisible by 2 there is no point in checking to see if it is divisible by any larger even num- bers. is suggests that the values used for test-divisor should not be 2, 3, 4, 5, 6, . . ., but rather 2, 3, 5, 7, 9, . . .. 71 To implement this change, deﬁne a procedure next that re- turns 3 if its input is equal to 2 and otherwise returns its in- put plus 2. Modify the smallest-divisor procedure to use (next test-divisor) instead of (+ test-divisor 1). With timed-prime-test incorporating this modiﬁed ver- sion of smallest-divisor, run the test for each of the 12 primes found in Exercise 1.22. Since this modiﬁcation halves the number of test steps, you should expect it to run about twice as fast. Is this expectation conﬁrmed? If not, what is the observed ratio of the speeds of the two algorithms, and how do you explain the fact that it is diﬀerent from 2? Exercise 1.24: Modify the timed-prime-test procedure of Exercise 1.22 to use fast-prime? (the Fermat method), and test each of the 12 primes you found in that exercise. Since the Fermat test has Θ(log n) growth, how would you expect the time to test primes near 1,000,000 to compare with the time needed to test primes near 1000? Do your data bear this out? Can you explain any discrepancy you ﬁnd? Exercise 1.25: Alyssa P. Hacker complains that we went to a lot of extra work in writing expmod. Aer all, she says, since we already know how to compute exponentials, we could have simply wrien (define (expmod base exp m) (remainder (fast-expt base exp) m)) Is she correct? Would this procedure serve as well for our fast prime tester? Explain. 72 Exercise 1.26: Louis Reasoner is having great diﬃculty do- ing Exercise 1.24. His fast-prime? test seems to run more slowly than his prime? test. Louis calls his friend Eva Lu Ator over to help. When they examine Louis’s code, they ﬁnd that he has rewrien the expmod procedure to use an explicit multiplication, rather than calling square: (define (expmod base exp m) (cond ((= exp 0) 1) ((even? exp) (remainder (* (expmod base (/ exp 2) m) (expmod base (/ exp 2) m)) m)) (else (remainder (* base (expmod base (- exp 1) m)) m)))) “I don’t see what diﬀerence that could make,” says Louis. “I do.” says Eva. “By writing the procedure like that, you have transformed the Θ(log n) process into a Θ(n) process.” Explain. Exercise 1.27: Demonstrate that the Carmichael numbers listed in Footnote 1.47 really do fool the Fermat test. at is, write a procedure that takes an integer n and tests whether an is congruent to a modulo n for every a < n, and try your procedure on the given Carmichael numbers. Exercise 1.28: One variant of the Fermat test that cannot be fooled is called the Miller-Rabin test (Miller 1976; Rabin 1980). is starts from an alternate form of Fermat’s Lile 73 eorem, which states that if n is a prime number and a is any positive integer less than n, then a raised to the (n−1)-st power is congruent to 1 modulo n. To test the primality of a number n by the Miller-Rabin test, we pick a random num- ber a < n and raise a to the (n − 1)-st power modulo n using the expmod procedure. However, whenever we perform the squaring step in expmod, we check to see if we have discov- ered a “nontrivial square root of 1 modulo n,” that is, a num- ber not equal to 1 or n −1 whose square is equal to 1 modulo n. It is possible to prove that if such a nontrivial square root of 1 exists, then n is not prime. It is also possible to prove that if n is an odd number that is not prime, then, for at least half the numbers a < n, computing an −1 in this way will reveal a nontrivial square root of 1 modulo n. (is is why the Miller-Rabin test cannot be fooled.) Modify the expmod procedure to signal if it discovers a nontrivial square root of 1, and use this to implement the Miller-Rabin test with a procedure analogous to fermat-test. Check your pro- cedure by testing various known primes and non-primes. Hint: One convenient way to make expmod signal is to have it return 0. 1.3 Formulating Abstractions with Higher-Order Procedures We have seen that procedures are, in eﬀect, abstractions that describe compound operations on numbers independent of the particular num- bers. For example, when we 74 (define (cube x) (* x x x)) we are not talking about the cube of a particular number, but rather about a method for obtaining the cube of any number. Of course we could get along without ever deﬁning this procedure, by always writing expressions such as (* 3 3 3) (* x x x) (* y y y) and never mentioning cube explicitly. is would place us at a serious disadvantage, forcing us to work always at the level of the particular op- erations that happen to be primitives in the language (multiplication, in this case) rather than in terms of higher-level operations. Our programs would be able to compute cubes, but our language would lack the ability to express the concept of cubing. One of the things we should demand from a powerful programming language is the ability to build abstrac- tions by assigning names to common paerns and then to work in terms of the abstractions directly. Procedures provide this ability. is is why all but the most primitive programming languages include mechanisms for deﬁning procedures. Yet even in numerical processing we will be severely limited in our ability to create abstractions if we are restricted to procedures whose pa- rameters must be numbers. Oen the same programming paern will be used with a number of diﬀerent procedures. To express such paerns as concepts, we will need to construct procedures that can accept pro- cedures as arguments or return procedures as values. Procedures that manipulate procedures are called higher-order procedures. is section shows how higher-order procedures can serve as powerful abstraction mechanisms, vastly increasing the expressive power of our language. 75 1.3.1 Procedures as Arguments Consider the following three procedures. e ﬁrst computes the sum of the integers from a through b: (define (sum-integers a b) (if (> a b) 0 (+ a (sum-integers (+ a 1) b)))) e second computes the sum of the cubes of the integers in the given range: (define (sum-cubes a b) (if (> a b) 0 (+ (cube a) (sum-cubes (+ a 1) b)))) e third computes the sum of a sequence of terms in the series 1 1 1 + + +..., 1 · 3 5 · 7 9 · 11 which converges to π /8 (very slowly):49 (define (pi-sum a b) (if (> a b) 0 (+ (/ 1.0 (* a (+ a 2))) (pi-sum (+ a 4) b)))) 49 is series, usually wrien in the equivalent form π4 = 1 − 13 + 15 − 17 + . . ., is due to Leibniz. We’ll see how to use this as the basis for some fancy numerical tricks in Section 3.5.3. 76 ese three procedures clearly share a common underlying paern. ey are for the most part identical, diﬀering only in the name of the procedure, the function of a used to compute the term to be added, and the function that provides the next value of a. We could generate each of the procedures by ﬁlling in slots in the same template: (define (⟨name⟩ a b) (if (> a b) 0 (+ (⟨term⟩ a) (⟨name⟩ (⟨next⟩ a) b)))) e presence of such a common paern is strong evidence that there is a useful abstraction waiting to be brought to the surface. Indeed, math- ematicians long ago identiﬁed the abstraction of summation of a series and invented “sigma notation,” for example ∑ b f (n) = f (a) + · · · + f (b), n=a to express this concept. e power of sigma notation is that it allows mathematicians to deal with the concept of summation itself rather than only with particular sums—for example, to formulate general results about sums that are independent of the particular series being summed. Similarly, as program designers, we would like our language to be powerful enough so that we can write a procedure that expresses the concept of summation itself rather than only procedures that compute particular sums. We can do so readily in our procedural language by taking the common template shown above and transforming the “slots” into formal parameters: (define (sum term a next b) (if (> a b) 77 0 (+ (term a) (sum term (next a) next b)))) Notice that sum takes as its arguments the lower and upper bounds a and b together with the procedures term and next. We can use sum just as we would any procedure. For example, we can use it (along with a procedure inc that increments its argument by 1) to deﬁne sum-cubes: (define (inc n) (+ n 1)) (define (sum-cubes a b) (sum cube a inc b)) Using this, we can compute the sum of the cubes of the integers from 1 to 10: (sum-cubes 1 10) 3025 With the aid of an identity procedure to compute the term, we can deﬁne sum-integers in terms of sum: (define (identity x) x) (define (sum-integers a b) (sum identity a inc b)) en we can add up the integers from 1 to 10: (sum-integers 1 10) 55 We can also deﬁne pi-sum in the same way:50 50 Notice that we have used block structure (Section 1.1.8) to embed the deﬁnitions of pi-next and pi-term within pi-sum, since these procedures are unlikely to be useful for any other purpose. We will see how to get rid of them altogether in Section 1.3.2. 78 (define (pi-sum a b) (define (pi-term x) (/ 1.0 (* x (+ x 2)))) (define (pi-next x) (+ x 4)) (sum pi-term a pi-next b)) Using these procedures, we can compute an approximation to π : (* 8 (pi-sum 1 1000)) 3.139592655589783 Once we have sum, we can use it as a building block in formulating fur- ther concepts. For instance, the deﬁnite integral of a function f between the limits a and b can be approximated numerically using the formula ∫ b [ ( ) ( ) ( ) ] dx dx dx f = f a+ + f a + dx + + f a + 2dx + + . . . dx a 2 2 2 for small values of dx. We can express this directly as a procedure: (define (integral f a b dx) (define (add-dx x) (+ x dx)) (* (sum f (+ a (/ dx 2.0)) add-dx b) dx)) (integral cube 0 1 0.01) .24998750000000042 (integral cube 0 1 0.001) .249999875000001 (e exact value of the integral of cube between 0 and 1 is 1/4.) 79 Exercise 1.29: Simpson’s Rule is a more accurate method of numerical integration than the method illustrated above. Using Simpson’s Rule, the integral of a function f between a and b is approximated as h (y0 + 4y1 + 2y2 + 4y3 + 2y4 + · · · + 2yn −2 + 4yn −1 + yn ), 3 where h = (b − a)/n, for some even integer n, and yk = f (a + kh). (Increasing n increases the accuracy of the ap- proximation.) Deﬁne a procedure that takes as arguments f , a, b, and n and returns the value of the integral, com- puted using Simpson’s Rule. Use your procedure to inte- grate cube between 0 and 1 (with n = 100 and n = 1000), and compare the results to those of the integral procedure shown above. Exercise 1.30: e sum procedure above generates a linear recursion. e procedure can be rewrien so that the sum is performed iteratively. Show how to do this by ﬁlling in the missing expressions in the following deﬁnition: (define (sum term a next b) (define (iter a result) (if ⟨??⟩ ⟨??⟩ (iter ⟨??⟩ ⟨??⟩))) (iter ⟨??⟩ ⟨??⟩)) Exercise 1.31: a. e sum procedure is only the simplest of a vast num- ber of similar abstractions that can be captured as higher- 80 order procedures.51 Write an analogous procedure called product that returns the product of the values of a function at points over a given range. Show how to de- ﬁne factorial in terms of product. Also use product to compute approximations to π using the formula52 π 2 · 4 · 4 · 6 · 6 · 8··· = . 4 3 · 3 · 5 · 5 · 7 · 7··· b. If your product procedure generates a recursive pro- cess, write one that generates an iterative process. If it generates an iterative process, write one that gen- erates a recursive process. Exercise 1.32: a. Show that sum and product (Exercise 1.31) are both special cases of a still more general notion called accumulate that combines a collection of terms, using some gen- eral accumulation function: (accumulate combiner null-value term a next b) 51 e intent of Exercise 1.31 through Exercise 1.33 is to demonstrate the expressive power that is aained by using an appropriate abstraction to consolidate many seem- ingly disparate operations. However, though accumulation and ﬁltering are elegant ideas, our hands are somewhat tied in using them at this point since we do not yet have data structures to provide suitable means of combination for these abstractions. We will return to these ideas in Section 2.2.3 when we show how to use sequences as interfaces for combining ﬁlters and accumulators to build even more powerful abstrac- tions. We will see there how these methods really come into their own as a powerful and elegant approach to designing programs. 52 is formula was discovered by the seventeenth-century English mathematician John Wallis. 81 accumulate takes as arguments the same term and range speciﬁcations as sum and product, together with a combiner procedure (of two arguments) that speci- ﬁes how the current term is to be combined with the accumulation of the preceding terms and a null-value that speciﬁes what base value to use when the terms run out. Write accumulate and show how sum and product can both be deﬁned as simple calls to accumulate. b. If your accumulate procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that gen- erates a recursive process. Exercise 1.33: You can obtain an even more general ver- sion of accumulate (Exercise 1.32) by introducing the no- tion of a ﬁlter on the terms to be combined. at is, combine only those terms derived from values in the range that sat- isfy a speciﬁed condition. e resulting filtered-accumulate abstraction takes the same arguments as accumulate, to- gether with an additional predicate of one argument that speciﬁes the ﬁlter. Write filtered-accumulate as a proce- dure. Show how to express the following using filtered- accumulate: a. the sum of the squares of the prime numbers in the interval a to b (assuming that you have a prime? pred- icate already wrien) b. the product of all the positive integers less than n that are relatively prime to n (i.e., all positive integers i < n such that (i, n) = 1). 82 1.3.2 Constructing Procedures Using lambda In using sum as in Section 1.3.1, it seems terribly awkward to have to deﬁne trivial procedures such as pi-term and pi-next just so we can use them as arguments to our higher-order procedure. Rather than de- ﬁne pi-next and pi-term, it would be more convenient to have a way to directly specify “the procedure that returns its input incremented by 4” and “the procedure that returns the reciprocal of its input times its input plus 2.” We can do this by introducing the special form lambda, which creates procedures. Using lambda we can describe what we want as (lambda (x) (+ x 4)) and (lambda (x) (/ 1.0 (* x (+ x 2)))) en our pi-sum procedure can be expressed without deﬁning any aux- iliary procedures as (define (pi-sum a b) (sum (lambda (x) (/ 1.0 (* x (+ x 2)))) a (lambda (x) (+ x 4)) b)) Again using lambda, we can write the integral procedure without hav- ing to deﬁne the auxiliary procedure add-dx: (define (integral f a b dx) (* (sum f (+ a (/ dx 2.0)) (lambda (x) (+ x dx)) b) dx)) 83 In general, lambda is used to create procedures in the same way as define, except that no name is speciﬁed for the procedure: (lambda (⟨formal-parameters⟩) ⟨body⟩) e resulting procedure is just as much a procedure as one that is cre- ated using define. e only diﬀerence is that it has not been associated with any name in the environment. In fact, (define (plus4 x) (+ x 4)) is equivalent to (define plus4 (lambda (x) (+ x 4))) We can read a lambda expression as follows: (lambda (x) (+ x 4)) | | | | | the procedure of an argument x that adds x and 4 Like any expression that has a procedure as its value, a lambda expres- sion can be used as the operator in a combination such as ((lambda (x y z) (+ x y (square z))) 1 2 3) 12 or, more generally, in any context where we would normally use a pro- cedure name.53 53 It would be clearer and less intimidating to people learning Lisp if a name more obvious than lambda, such as make-procedure, were used. But the convention is ﬁrmly entrenched. e notation is adopted from the λ-calculus, a mathematical formalism in- troduced by the mathematical logician Alonzo Church (1941). Church developed the λ-calculus to provide a rigorous foundation for studying the notions of function and function application. e λ-calculus has become a basic tool for mathematical investi- gations of the semantics of programming languages. 84 Using let to create local variables Another use of lambda is in creating local variables. We oen need lo- cal variables in our procedures other than those that have been bound as formal parameters. For example, suppose we wish to compute the function f (x , y) = x(1 + xy)2 + y(1 − y) + (1 + xy)(1 − y), which we could also express as a = 1 + xy, b = 1 − y, f (x , y) = xa 2 + yb + ab. In writing a procedure to compute f , we would like to include as local variables not only x and y but also the names of intermediate quantities like a and b. One way to accomplish this is to use an auxiliary procedure to bind the local variables: (define (f x y) (define (f-helper a b) (+ (* x (square a)) (* y b) (* a b))) (f-helper (+ 1 (* x y)) (- 1 y))) Of course, we could use a lambda expression to specify an anonymous procedure for binding our local variables. e body of f then becomes a single call to that procedure: (define (f x y) ((lambda (a b) 85 (+ (* x (square a)) (* y b) (* a b))) (+ 1 (* x y)) (- 1 y))) is construct is so useful that there is a special form called let to make its use more convenient. Using let, the f procedure could be wrien as (define (f x y) (let ((a (+ 1 (* x y))) (b (- 1 y))) (+ (* x (square a)) (* y b) (* a b)))) e general form of a let expression is (let ((⟨var1 ⟩ ⟨exp1 ⟩) (⟨var2 ⟩ ⟨exp2 ⟩) ... (⟨varn ⟩ ⟨expn ⟩)) ⟨body⟩) which can be thought of as saying let⟨var1 ⟩ have the value ⟨exp1 ⟩ and ⟨var2 ⟩ have the value ⟨exp2 ⟩ and ... ⟨varn ⟩ have the value ⟨expn ⟩ in ⟨body⟩ e ﬁrst part of the let expression is a list of name-expression pairs. When the let is evaluated, each name is associated with the value of the corresponding expression. e body of the let is evaluated with these names bound as local variables. e way this happens is that the let expression is interpreted as an alternate syntax for 86 ((lambda (⟨var1 ⟩ . . . ⟨varn ⟩) ⟨body⟩) ⟨exp1 ⟩ ... ⟨expn ⟩) No new mechanism is required in the interpreter in order to provide local variables. A let expression is simply syntactic sugar for the un- derlying lambda application. We can see from this equivalence that the scope of a variable spec- iﬁed by a let expression is the body of the let. is implies that: • let allows one to bind variables as locally as possible to where they are to be used. For example, if the value of x is 5, the value of the expression (+ (let ((x 3)) (+ x (* x 10))) x) is 38. Here, the x in the body of the let is 3, so the value of the let expression is 33. On the other hand, the x that is the second argument to the outermost + is still 5. • e variables’ values are computed outside the let. is maers when the expressions that provide the values for the local vari- ables depend upon variables having the same names as the local variables themselves. For example, if the value of x is 2, the ex- pression (let ((x 3) (y (+ x 2))) (* x y)) 87 will have the value 12 because, inside the body of the let, x will be 3 and y will be 4 (which is the outer x plus 2). Sometimes we can use internal deﬁnitions to get the same eﬀect as with let. For example, we could have deﬁned the procedure f above as (define (f x y) (define a (+ 1 (* x y))) (define b (- 1 y)) (+ (* x (square a)) (* y b) (* a b))) We prefer, however, to use let in situations like this and to use internal define only for internal procedures.54 Exercise 1.34: Suppose we deﬁne the procedure (define (f g) (g 2)) en we have (f square) 4 (f (lambda (z) (* z (+ z 1)))) 6 What happens if we (perversely) ask the interpreter to eval- uate the combination (f f)? Explain. 54 Understanding internal deﬁnitions well enough to be sure a program means what we intend it to mean requires a more elaborate model of the evaluation process than we have presented in this chapter. e subtleties do not arise with internal deﬁnitions of procedures, however. We will return to this issue in Section 4.1.6, aer we learn more about evaluation. 88 1.3.3 Procedures as General Methods We introduced compound procedures in Section 1.1.4 as a mechanism for abstracting paerns of numerical operations so as to make them in- dependent of the particular numbers involved. With higher-order pro- cedures, such as the integral procedure of Section 1.3.1, we began to see a more powerful kind of abstraction: procedures used to express general methods of computation, independent of the particular func- tions involved. In this section we discuss two more elaborate examples— general methods for ﬁnding zeros and ﬁxed points of functions—and show how these methods can be expressed directly as procedures. Finding roots of equations by the half-interval method e half-interval method is a simple but powerful technique for ﬁnding roots of an equation f (x) = 0, where f is a continuous function. e idea is that, if we are given points a and b such that f (a) < 0 < f (b), then f must have at least one zero between a and b. To locate a zero, let x be the average of a and b, and compute f (x ). If f (x ) > 0, then f must have a zero between a and x. If f (x) < 0, then f must have a zero between x and b. Continuing in this way, we can identify smaller and smaller intervals on which f must have a zero. When we reach a point where the interval is small enough, the process stops. Since the interval of uncertainty is reduced by half at each step of the process, the number of steps required grows as Θ(log(L/T )), where L is the length of the original interval and T is the error tolerance (that is, the size of the interval we will consider “small enough”). Here is a procedure that implements this strategy: (define (search f neg-point pos-point) (let ((midpoint (average neg-point pos-point))) 89 (if (close-enough? neg-point pos-point) midpoint (let ((test-value (f midpoint))) (cond ((positive? test-value) (search f neg-point midpoint)) ((negative? test-value) (search f midpoint pos-point)) (else midpoint)))))) We assume that we are initially given the function f together with points at which its values are negative and positive. We ﬁrst compute the midpoint of the two given points. Next we check to see if the given interval is small enough, and if so we simply return the midpoint as our answer. Otherwise, we compute as a test value the value of f at the mid- point. If the test value is positive, then we continue the process with a new interval running from the original negative point to the midpoint. If the test value is negative, we continue with the interval from the mid- point to the positive point. Finally, there is the possibility that the test value is 0, in which case the midpoint is itself the root we are searching for. To test whether the endpoints are “close enough” we can use a pro- cedure similar to the one used in Section 1.1.7 for computing square roots:55 (define (close-enough? x y) (< (abs (- x y)) 0.001)) search is awkward to use directly, because we can accidentally give it points at which f ’s values do not have the required sign, in which case 55 We have used 0.001 as a representative “small” number to indicate a tolerance for the acceptable error in a calculation. e appropriate tolerance for a real calculation depends upon the problem to be solved and the limitations of the computer and the algorithm. is is oen a very subtle consideration, requiring help from a numerical analyst or some other kind of magician. 90 we get a wrong answer. Instead we will use search via the following procedure, which checks to see which of the endpoints has a negative function value and which has a positive value, and calls the search pro- cedure accordingly. If the function has the same sign on the two given points, the half-interval method cannot be used, in which case the pro- cedure signals an error.56 (define (half-interval-method f a b) (let ((a-value (f a)) (b-value (f b))) (cond ((and (negative? a-value) (positive? b-value)) (search f a b)) ((and (negative? b-value) (positive? a-value)) (search f b a)) (else (error "Values are not of opposite sign" a b))))) e following example uses the half-interval method to approximate π as the root between 2 and 4 of sin x = 0: (half-interval-method sin 2.0 4.0) 3.14111328125 Here is another example, using the half-interval method to search for a root of the equation x 3 − 2x − 3 = 0 between 1 and 2: (half-interval-method (lambda (x) (- (* x x x) (* 2 x) 3)) 1.0 2.0) 1.89306640625 56 is can be accomplished using error, which takes as arguments a number of items that are printed as error messages. 91 Finding fixed points of functions A number x is called a ﬁxed point of a function f if x satisﬁes the equa- tion f (x) = x. For some functions f we can locate a ﬁxed point by beginning with an initial guess and applying f repeatedly, f (x), f (f (x)), f (f (f (x))), ..., until the value does not change very much. Using this idea, we can de- vise a procedure fixed-point that takes as inputs a function and an initial guess and produces an approximation to a ﬁxed point of the func- tion. We apply the function repeatedly until we ﬁnd two successive val- ues whose diﬀerence is less than some prescribed tolerance: (define tolerance 0.00001) (define (fixed-point f first-guess) (define (close-enough? v1 v2) (< (abs (- v1 v2)) tolerance)) (define (try guess) (let ((next (f guess))) (if (close-enough? guess next) next (try next)))) (try first-guess)) For example, we can use this method to approximate the ﬁxed point of the cosine function, starting with 1 as an initial approximation:57 (fixed-point cos 1.0) .7390822985224023 Similarly, we can ﬁnd a solution to the equation y = sin y + cos y: 57 Try this during a boring lecture: Set your calculator to radians mode and then repeatedly press the cos buon until you obtain the ﬁxed point. 92 (fixed-point (lambda (y) (+ (sin y) (cos y))) 1.0) 1.2587315962971173 e ﬁxed-point process is reminiscent of the process we used for ﬁnding square roots in Section 1.1.7. Both are based on the idea of repeatedly improving a guess until the result satisﬁes some criterion. In fact, we can readily formulate the square-root computation as a ﬁxed-point search. Computing the square root of some number x requires ﬁnding a y such that y 2 = x. Puing this equation into the equivalent form y = x/y, we recognize that we are looking for a ﬁxed point of the function58 y 7→ x/y, and we can therefore try to compute square roots as (define (sqrt x) (fixed-point (lambda (y) (/ x y)) 1.0)) Unfortunately, this ﬁxed-point search does not converge. Consider an initial guess y1 . e next guess is y2 = x/y1 and the next guess is y3 = x/y2 = x/(x/y1 ) = y1 . is results in an inﬁnite loop in which the two guesses y1 and y2 repeat over and over, oscillating about the answer. One way to control such oscillations is to prevent the guesses from changing so much. Since the answer is always between our guess y and x/y, we can make a new guess that is not as far from y as x/y by av- eraging y with x/y, so that the next guess aer y is 21 (y + x/y) instead of x/y. e process of making such a sequence of guesses is simply the process of looking for a ﬁxed point of y 7→ 12 (y + x/y): (define (sqrt x) (fixed-point (lambda (y) (average y (/ x y))) 1.0)) 58 7→ (pronounced “maps to”) is the mathematician’s way of writing lambda. y 7→ x/y means (lambda (y) (/ x y)), that is, the function whose value at y is x/y. 93 (Note that y = 12 (y + x/y) is a simple transformation of the equation y = x/y; to derive it, add y to both sides of the equation and divide by 2.) With this modiﬁcation, the square-root procedure works. In fact, if we unravel the deﬁnitions, we can see that the sequence of approxi- mations to the square root generated here is precisely the same as the one generated by our original square-root procedure of Section 1.1.7. is approach of averaging successive approximations to a solution, a technique that we call average damping, oen aids the convergence of ﬁxed-point searches. Exercise 1.35: Show that the golden ratio ϕ (Section 1.2.2) is a ﬁxed point of the transformation x 7→ 1 + 1/x , and use this fact to compute ϕ by means of the fixed-point procedure. Exercise 1.36: Modify fixed-point so that it prints the sequence of approximations it generates, using the newline and display primitives shown in Exercise 1.22. en ﬁnd a solution to x x = 1000 by ﬁnding a ﬁxed point of x 7→ log(1000)/ log(x). (Use Scheme’s primitive log procedure, which computes natural logarithms.) Compare the number of steps this takes with and without average damping. (Note that you cannot start fixed-point with a guess of 1, as this would cause division by log(1) = 0.) Exercise 1.37: a. An inﬁnite continued fraction is an expression of the 94 form N1 f = . N2 D1 + N3 D2 + D3 + . . . As an example, one can show that the inﬁnite con- tinued fraction expansion with the Ni and the Di all equal to 1 produces 1/ϕ, where ϕ is the golden ratio (described in Section 1.2.2). One way to approximate an inﬁnite continued fraction is to truncate the expan- sion aer a given number of terms. Such a truncation— a so-called k-term ﬁnite continued fraction—has the form N1 . N2 D1 + .. Nk .+ Dk Suppose that n and d are procedures of one argument (the term index i) that return the Ni and Di of the terms of the continued fraction. Deﬁne a procedure cont-frac such that evaluating (cont-frac n d k) computes the value of the k-term ﬁnite continued frac- tion. Check your procedure by approximating 1/ϕ us- ing (cont-frac (lambda (i) 1.0) (lambda (i) 1.0) k) 95 for successive values of k. How large must you make k in order to get an approximation that is accurate to 4 decimal places? b. If your cont-frac procedure generates a recursive pro- cess, write one that generates an iterative process. If it generates an iterative process, write one that gen- erates a recursive process. Exercise 1.38: In 1737, the Swiss mathematician Leonhard Euler published a memoir De Fractionibus Continuis, which included a continued fraction expansion for e − 2, where e is the base of the natural logarithms. In this fraction, the Ni are all 1, and the Di are successively 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, . . .. Write a program that uses your cont-frac procedure from Exercise 1.37 to approximate e, based on Euler’s expansion. Exercise 1.39: A continued fraction representation of the tangent function was published in 1770 by the German math- ematician J.H. Lambert: x tan x = , x2 1− x2 3− 5−... where x is in radians. Deﬁne a procedure (tan-cf x k) that computes an approximation to the tangent function based on Lambert’s formula. k speciﬁes the number of terms to compute, as in Exercise 1.37. 96 1.3.4 Procedures as Returned Values e above examples demonstrate how the ability to pass procedures as arguments signiﬁcantly enhances the expressive power of our program- ming language. We can achieve even more expressive power by creating procedures whose returned values are themselves procedures. We can illustrate this idea by looking again at the ﬁxed-point exam- ple described at the end of Section 1.3.3. We formulated a new version of the square-root procedure as a ﬁxed-point search, starting with the √ observation that x is a ﬁxed-point of the function y 7→ x/y. en we used average damping to make the approximations converge. Average damping is a useful general technique in itself. Namely, given a function f , we consider the function whose value at x is equal to the average of x and f (x ). We can express the idea of average damping by means of the fol- lowing procedure: (define (average-damp f) (lambda (x) (average x (f x)))) average-damp is a procedure that takes as its argument a procedure f and returns as its value a procedure (produced by the lambda) that, when applied to a number x, produces the average of x and (f x). For example, applying average-damp to the square procedure produces a procedure whose value at some number x is the average of x and x 2 . Applying this resulting procedure to 10 returns the average of 10 and 100, or 55:59 59 Observe that this is a combination whose operator is itself a combination. Exercise 1.4 already demonstrated the ability to form such combinations, but that was only a toy example. Here we begin to see the real need for such combinations—when applying a procedure that is obtained as the value returned by a higher-order procedure. 97 ((average-damp square) 10) 55 Using average-damp, we can reformulate the square-root procedure as follows: (define (sqrt x) (fixed-point (average-damp (lambda (y) (/ x y))) 1.0)) Notice how this formulation makes explicit the three ideas in the method: ﬁxed-point search, average damping, and the function y 7→ x/y. It is in- structive to compare this formulation of the square-root method with the original version given in Section 1.1.7. Bear in mind that these pro- cedures express the same process, and notice how much clearer the idea becomes when we express the process in terms of these abstractions. In general, there are many ways to formulate a process as a procedure. Ex- perienced programmers know how to choose procedural formulations that are particularly perspicuous, and where useful elements of the pro- cess are exposed as separate entities that can be reused in other appli- cations. As a simple example of reuse, notice that the cube root of x is a ﬁxed point of the function y 7→ x/y 2 , so we can immediately generalize our square-root procedure to one that extracts cube roots:60 (define (cube-root x) (fixed-point (average-damp (lambda (y) (/ x (square y)))) 1.0)) Newton’s method When we ﬁrst introduced the square-root procedure, in Section 1.1.7, we mentioned that this was a special case of Newton’s method. If x 7→ д(x ) 60 See Exercise 1.45 for a further generalization. 98 is a diﬀerentiable function, then a solution of the equation д(x) = 0 is a ﬁxed point of the function x 7→ f (x ), where д(x) f (x ) = x − Dд(x) and Dд(x) is the derivative of д evaluated at x. Newton’s method is the use of the ﬁxed-point method we saw above to approximate a solution of the equation by ﬁnding a ﬁxed point of the function f.61 For many functions д and for suﬃciently good initial guesses for x , Newton’s method converges very rapidly to a solution of д(x ) = 0.62 In order to implement Newton’s method as a procedure, we must ﬁrst express the idea of derivative. Note that “derivative,” like average damping, is something that transforms a function into another function. For instance, the derivative of the function x 7→ x 3 is the function x 7→ 3x 2. In general, if д is a function and dx is a small number, then the derivative Dд of д is the function whose value at any number x is given (in the limit of small dx) by д(x + dx) − д(x) Dд(x) = . dx us, we can express the idea of derivative (taking dx to be, say, 0.00001) as the procedure (define (deriv g) (lambda (x) (/ (- (g (+ x dx)) (g x)) dx))) 61 Elementary calculus books usually describe Newton’s method in terms of the se- quence of approximations xn+1 = xn −д(xn )/Dд(xn ). Having language for talking about processes and using the idea of ﬁxed points simpliﬁes the description of the method. 62 Newton’s method does not always converge to an answer, but it can be shown that in favorable cases each iteration doubles the number-of-digits accuracy of the ap- proximation to the solution. In such cases, Newton’s method will converge much more rapidly than the half-interval method. 99 along with the deﬁnition (define dx 0.00001) Like average-damp, deriv is a procedure that takes a procedure as ar- gument and returns a procedure as value. For example, to approximate the derivative of x 7→ x 3 at 5 (whose exact value is 75) we can evaluate (define (cube x) (* x x x)) ((deriv cube) 5) 75.00014999664018 With the aid of deriv, we can express Newton’s method as a ﬁxed-point process: (define (newton-transform g) (lambda (x) (- x (/ (g x) ((deriv g) x))))) (define (newtons-method g guess) (fixed-point (newton-transform g) guess)) e newton-transform procedure expresses the formula at the begin- ning of this section, and newtons-method is readily deﬁned in terms of this. It takes as arguments a procedure that computes the function for which we want to ﬁnd a zero, together with an initial guess. For in- stance, to ﬁnd the square root of x, we can use Newton’s method to ﬁnd a zero of the function y 7→ y 2 − x starting with an initial guess of 1.63 is provides yet another form of the square-root procedure: (define (sqrt x) (newtons-method (lambda (y) (- (square y) x)) 1.0)) 63 Forﬁnding square roots, Newton’s method converges rapidly to the correct solu- tion from any starting point. 100 Abstractions and first-class procedures We’ve seen two ways to express the square-root computation as an in- stance of a more general method, once as a ﬁxed-point search and once using Newton’s method. Since Newton’s method was itself expressed as a ﬁxed-point process, we actually saw two ways to compute square roots as ﬁxed points. Each method begins with a function and ﬁnds a ﬁxed point of some transformation of the function. We can express this general idea itself as a procedure: (define (fixed-point-of-transform g transform guess) (fixed-point (transform g) guess)) is very general procedure takes as its arguments a procedure g that computes some function, a procedure that transforms g, and an initial guess. e returned result is a ﬁxed point of the transformed function. Using this abstraction, we can recast the ﬁrst square-root computa- tion from this section (where we look for a ﬁxed point of the average- damped version of y 7→ x/y) as an instance of this general method: (define (sqrt x) (fixed-point-of-transform (lambda (y) (/ x y)) average-damp 1.0)) Similarly, we can express the second square-root computation from this section (an instance of Newton’s method that ﬁnds a ﬁxed point of the Newton transform of y 7→ y 2 − x) as (define (sqrt x) (fixed-point-of-transform (lambda (y) (- (square y) x)) newton-transform 1.0)) We began Section 1.3 with the observation that compound procedures are a crucial abstraction mechanism, because they permit us to express general methods of computing as explicit elements in our programming 101 language. Now we’ve seen how higher-order procedures permit us to manipulate these general methods to create further abstractions. As programmers, we should be alert to opportunities to identify the underlying abstractions in our programs and to build upon them and generalize them to create more powerful abstractions. is is not to say that one should always write programs in the most abstract way possible; expert programmers know how to choose the level of abstrac- tion appropriate to their task. But it is important to be able to think in terms of these abstractions, so that we can be ready to apply them in new contexts. e signiﬁcance of higher-order procedures is that they enable us to represent these abstractions explicitly as elements in our programming language, so that they can be handled just like other com- putational elements. In general, programming languages impose restrictions on the ways in which computational elements can be manipulated. Elements with the fewest restrictions are said to have ﬁrst-class status. Some of the “rights and privileges” of ﬁrst-class elements are:64 • ey may be named by variables. • ey may be passed as arguments to procedures. • ey may be returned as the results of procedures. • ey may be included in data structures.65 Lisp, unlike other common programming languages, awards procedures full ﬁrst-class status. is poses challenges for eﬃcient implementation, 64 e notion of ﬁrst-class status of programming-language elements is due to the British computer scientist Christopher Strachey (1916-1975). 65 We’ll see examples of this aer we introduce data structures in Chapter 2. 102 but the resulting gain in expressive power is enormous.66 Exercise 1.40: Deﬁne a procedure cubic that can be used together with the newtons-method procedure in expressions of the form (newtons-method (cubic a b c) 1) to approximate zeros of the cubic x 3 + ax 2 + bx + c. Exercise 1.41: Deﬁne a procedure double that takes a pro- cedure of one argument as argument and returns a proce- dure that applies the original procedure twice. For exam- ple, if inc is a procedure that adds 1 to its argument, then (double inc) should be a procedure that adds 2. What value is returned by (((double (double double)) inc) 5) Exercise 1.42: Let f and д be two one-argument functions. e composition f aer д is deﬁned to be the function x 7→ f (д(x)). Deﬁne a procedure compose that implements com- position. For example, if inc is a procedure that adds 1 to its argument, ((compose square inc) 6) 49 66 e major implementation cost of ﬁrst-class procedures is that allowing procedures to be returned as values requires reserving storage for a procedure’s free variables even while the procedure is not executing. In the Scheme implementation we will study in Section 4.1, these variables are stored in the procedure’s environment. 103 Exercise 1.43: If f is a numerical function and n is a posi- tive integer, then we can form the n th repeated application of f , which is deﬁned to be the function whose value at x is f (f (. . . (f (x )) . . . )). For example, if f is the function x 7→ x + 1, then the n th repeated application of f is the function x 7→ x +n. If f is the operation of squaring a num- ber, then the n th repeated application of f is the function that raises its argument to the 2n -th power. Write a proce- dure that takes as inputs a procedure that computes f and a positive integer n and returns the procedure that computes the n th repeated application of f . Your procedure should be able to be used as follows: ((repeated square 2) 5) 625 Hint: You may ﬁnd it convenient to use compose from Ex- ercise 1.42. Exercise 1.44: e idea of smoothing a function is an im- portant concept in signal processing. If f is a function and dx is some small number, then the smoothed version of f is the function whose value at a point x is the average of f (x − dx), f (x), and f (x +dx). Write a procedure smooth that takes as input a procedure that computes f and returns a proce- dure that computes the smoothed f . It is sometimes valu- able to repeatedly smooth a function (that is, smooth the smoothed function, and so on) to obtain the n-fold smoothed function. Show how to generate the n-fold smoothed func- tion of any given function using smooth and repeated from Exercise 1.43. 104 Exercise 1.45: We saw in Section 1.3.3 that aempting to compute square roots by naively ﬁnding a ﬁxed point of y 7→ x/y does not converge, and that this can be ﬁxed by average damping. e same method works for ﬁnding cube roots as ﬁxed points of the average-damped y 7→ x/y 2 . Un- fortunately, the process does not work for fourth roots—a single average damp is not enough to make a ﬁxed-point search for y 7→ x/y 3 converge. On the other hand, if we average damp twice (i.e., use the average damp of the av- erage damp of y 7→ x/y 3 ) the ﬁxed-point search does con- verge. Do some experiments to determine how many av- erage damps are required to compute n th roots as a ﬁxed- point search based upon repeated average damping of y 7→ x/yn −1 . Use this to implement a simple procedure for com- puting n th roots using fixed-point, average-damp, and the repeated procedure of Exercise 1.43. Assume that any arith- metic operations you need are available as primitives. Exercise 1.46: Several of the numerical methods described in this chapter are instances of an extremely general com- putational strategy known as iterative improvement. Itera- tive improvement says that, to compute something, we start with an initial guess for the answer, test if the guess is good enough, and otherwise improve the guess and continue the process using the improved guess as the new guess. Write a procedure iterative-improve that takes two procedures as arguments: a method for telling whether a guess is good enough and a method for improving a guess. iterative- improve should return as its value a procedure that takes a guess as argument and keeps improving the guess until it is 105 good enough. Rewrite the sqrt procedure of Section 1.1.7 and the fixed-point procedure of Section 1.3.3 in terms of iterative-improve. 106 Building Abstractions with Data We now come to the decisive step of mathematical abstrac- tion: we forget about what the symbols stand for. . . .[e mathematician] need not be idle; there are many operations which he may carry out with these symbols, without ever having to look at the things they stand for. —Hermann Weyl, e Mathematical Way of inking W Chapter 1 on computational processes and on the role of procedures in program design. We saw how to use primitive data (numbers) and primitive operations (arithmetic op- erations), how to combine procedures to form compound procedures through composition, conditionals, and the use of parameters, and how to abstract procedures by using define. We saw that a procedure can be regarded as a paern for the local evolution of a process, and we classiﬁed, reasoned about, and performed simple algorithmic analyses of some common paerns for processes as embodied in procedures. We 107 also saw that higher-order procedures enhance the power of our lan- guage by enabling us to manipulate, and thereby to reason in terms of, general methods of computation. is is much of the essence of pro- gramming. In this chapter we are going to look at more complex data. All the procedures in chapter 1 operate on simple numerical data, and simple data are not suﬃcient for many of the problems we wish to address using computation. Programs are typically designed to model complex phenomena, and more oen than not one must construct computational objects that have several parts in order to model real-world phenom- ena that have several aspects. us, whereas our focus in chapter 1 was on building abstractions by combining procedures to form compound procedures, we turn in this chapter to another key aspect of any pro- gramming language: the means it provides for building abstractions by combining data objects to form compound data. Why do we want compound data in a programming language? For the same reasons that we want compound procedures: to elevate the conceptual level at which we can design our programs, to increase the modularity of our designs, and to enhance the expressive power of our language. Just as the ability to deﬁne procedures enables us to deal with processes at a higher conceptual level than that of the primitive oper- ations of the language, the ability to construct compound data objects enables us to deal with data at a higher conceptual level than that of the primitive data objects of the language. Consider the task of designing a system to perform arithmetic with rational numbers. We could imagine an operation add-rat that takes two rational numbers and produces their sum. In terms of simple data, a rational number can be thought of as two integers: a numerator and a denominator. us, we could design a program in which each ratio- nal number would be represented by two integers (a numerator and a 108 denominator) and where add-rat would be implemented by two proce- dures (one producing the numerator of the sum and one producing the denominator). But this would be awkward, because we would then need to explicitly keep track of which numerators corresponded to which de- nominators. In a system intended to perform many operations on many rational numbers, such bookkeeping details would cluer the programs substantially, to say nothing of what they would do to our minds. It would be much beer if we could “glue together” a numerator and de- nominator to form a pair—a compound data object —that our programs could manipulate in a way that would be consistent with regarding a rational number as a single conceptual unit. e use of compound data also enables us to increase the modular- ity of our programs. If we can manipulate rational numbers directly as objects in their own right, then we can separate the part of our program that deals with rational numbers per se from the details of how rational numbers may be represented as pairs of integers. e general technique of isolating the parts of a program that deal with how data objects are represented from the parts of a program that deal with how data objects are used is a powerful design methodology called data abstraction. We will see how data abstraction makes programs much easier to design, maintain, and modify. e use of compound data leads to a real increase in the expressive power of our programming language. Consider the idea of forming a “linear combination” ax + by. We might like to write a procedure that would accept a, b, x, and y as arguments and return the value of ax +by. is presents no diﬃculty if the arguments are to be numbers, because we can readily deﬁne the procedure (define (linear-combination a b x y) (+ (* a x) (* b y))) 109 But suppose we are not concerned only with numbers. Suppose we would like to express, in procedural terms, the idea that one can form linear combinations whenever addition and multiplication are deﬁned— for rational numbers, complex numbers, polynomials, or whatever. We could express this as a procedure of the form (define (linear-combination a b x y) (add (mul a x) (mul b y))) where add and mul are not the primitive procedures + and * but rather more complex things that will perform the appropriate operations for whatever kinds of data we pass in as the arguments a, b, x, and y. e key point is that the only thing linear-combination should need to know about a, b, x, and y is that the procedures add and mul will per- form the appropriate manipulations. From the perspective of the pro- cedure linear-combination, it is irrelevant what a, b, x, and y are and even more irrelevant how they might happen to be represented in terms of more primitive data. is same example shows why it is important that our programming language provide the ability to manipulate com- pound objects directly: Without this, there is no way for a procedure such as linear-combination to pass its arguments along to add and mul without having to know their detailed structure.1 1 e ability to directly manipulate procedures provides an analogous increase in the expressive power of a programming language. For example, in Section 1.3.1 we intro- duced the sum procedure, which takes a procedure term as an argument and computes the sum of the values of term over some speciﬁed interval. In order to deﬁne sum, it is crucial that we be able to speak of a procedure such as term as an entity in its own right, without regard for how term might be expressed with more primitive operations. Indeed, if we did not have the notion of “a procedure,” it is doubtful that we would ever even think of the possibility of deﬁning an operation such as sum. Moreover, insofar as performing the summation is concerned, the details of how term may be constructed from more primitive operations are irrelevant. 110 We begin this chapter by implementing the rational-number arith- metic system mentioned above. is will form the background for our discussion of compound data and data abstraction. As with compound procedures, the main issue to be addressed is that of abstraction as a technique for coping with complexity, and we will see how data ab- straction enables us to erect suitable abstraction barriers between diﬀer- ent parts of a program. We will see that the key to forming compound data is that a pro- gramming language should provide some kind of “glue” so that data objects can be combined to form more complex data objects. ere are many possible kinds of glue. Indeed, we will discover how to form com- pound data using no special “data” operations at all, only procedures. is will further blur the distinction between “procedure” and “data,” which was already becoming tenuous toward the end of chapter 1. We will also explore some conventional techniques for representing sequences and trees. One key idea in dealing with compound data is the notion of closure—that the glue we use for combining data objects should allow us to combine not only primitive data objects, but compound data ob- jects as well. Another key idea is that compound data objects can serve as conventional interfaces for combining program modules in mix-and- match ways. We illustrate some of these ideas by presenting a simple graphics language that exploits closure. We will then augment the representational power of our language by introducing symbolic expressions—data whose elementary parts can be arbitrary symbols rather than only numbers. We explore various al- ternatives for representing sets of objects. We will ﬁnd that, just as a given numerical function can be computed by many diﬀerent computa- tional processes, there are many ways in which a given data structure can be represented in terms of simpler objects, and the choice of rep- resentation can have signiﬁcant impact on the time and space require- 111 ments of processes that manipulate the data. We will investigate these ideas in the context of symbolic diﬀerentiation, the representation of sets, and the encoding of information. Next we will take up the problem of working with data that may be represented diﬀerently by diﬀerent parts of a program. is leads to the need to implement generic operations, which must handle many diﬀerent types of data. Maintaining modularity in the presence of generic oper- ations requires more powerful abstraction barriers than can be erected with simple data abstraction alone. In particular, we introduce data- directed programming as a technique that allows individual data repre- sentations to be designed in isolation and then combined additively (i.e., without modiﬁcation). To illustrate the power of this approach to sys- tem design, we close the chapter by applying what we have learned to the implementation of a package for performing symbolic arithmetic on polynomials, in which the coeﬃcients of the polynomials can be inte- gers, rational numbers, complex numbers, and even other polynomials. 2.1 Introduction to Data Abstraction In Section 1.1.8, we noted that a procedure used as an element in creat- ing a more complex procedure could be regarded not only as a collection of particular operations but also as a procedural abstraction. at is, the details of how the procedure was implemented could be suppressed, and the particular procedure itself could be replaced by any other pro- cedure with the same overall behavior. In other words, we could make an abstraction that would separate the way the procedure would be used from the details of how the procedure would be implemented in terms of more primitive procedures. e analogous notion for compound data is called data abstraction. Data abstraction is a methodology that enables 112 us to isolate how a compound data object is used from the details of how it is constructed from more primitive data objects. e basic idea of data abstraction is to structure the programs that are to use compound data objects so that they operate on “abstract data.” at is, our programs should use data in such a way as to make no as- sumptions about the data that are not strictly necessary for performing the task at hand. At the same time, a “concrete” data representation is deﬁned independent of the programs that use the data. e interface be- tween these two parts of our system will be a set of procedures, called se- lectors and constructors, that implement the abstract data in terms of the concrete representation. To illustrate this technique, we will consider how to design a set of procedures for manipulating rational numbers. 2.1.1 Example: Arithmetic Operations for Rational Numbers Suppose we want to do arithmetic with rational numbers. We want to be able to add, subtract, multiply, and divide them and to test whether two rational numbers are equal. Let us begin by assuming that we already have a way of construct- ing a rational number from a numerator and a denominator. We also assume that, given a rational number, we have a way of extracting (or selecting) its numerator and its denominator. Let us further assume that the constructor and selectors are available as procedures: • (make-rat ⟨n⟩ ⟨d ⟩) returns the rational number whose numera- tor is the integer ⟨n⟩ and whose denominator is the integer ⟨d ⟩. • (numer ⟨x ⟩) returns the numerator of the rational number ⟨x⟩. • (denom ⟨x ⟩) returns the denominator of the rational number ⟨x ⟩. 113 We are using here a powerful strategy of synthesis: wishful thinking. We haven’t yet said how a rational number is represented, or how the procedures numer, denom, and make-rat should be implemented. Even so, if we did have these three procedures, we could then add, subtract, multiply, divide, and test equality by using the following relations: n1 n2 n 1d 2 + n 2d 1 + = , d1 d2 d 1d 2 n1 n2 n 1d 2 − n 2d 1 − = , d1 d2 d 1d 2 n1 n2 n 1n 2 · = , d1 d2 d 1d 2 n 1 /d 1 n 1d 2 = , n 2 /d 2 d 1n 2 n1 n2 = if and only if n 1d 2 = n 2d 1 . d1 d2 We can express these rules as procedures: (define (add-rat x y) (make-rat (+ (* (numer x) (denom y)) (* (numer y) (denom x))) (* (denom x) (denom y)))) (define (sub-rat x y) (make-rat (- (* (numer x) (denom y)) (* (numer y) (denom x))) (* (denom x) (denom y)))) (define (mul-rat x y) (make-rat (* (numer x) (numer y)) (* (denom x) (denom y)))) (define (div-rat x y) (make-rat (* (numer x) (denom y)) (* (denom x) (numer y)))) 114 (define (equal-rat? x y) (= (* (numer x) (denom y)) (* (numer y) (denom x)))) Now we have the operations on rational numbers deﬁned in terms of the selector and constructor procedures numer, denom, and make-rat. But we haven’t yet deﬁned these. What we need is some way to glue together a numerator and a denominator to form a rational number. Pairs To enable us to implement the concrete level of our data abstraction, our language provides a compound structure called a pair, which can be constructed with the primitive procedure cons. is procedure takes two arguments and returns a compound data object that contains the two arguments as parts. Given a pair, we can extract the parts using the primitive procedures car and cdr.2 us, we can use cons, car, and cdr as follows: (define x (cons 1 2)) (car x) 1 (cdr x) 2 Notice that a pair is a data object that can be given a name and manip- ulated, just like a primitive data object. Moreover, cons can be used to form pairs whose elements are pairs, and so on: 2 e name cons stands for “construct.” e names car and cdr derive from the orig- inal implementation of Lisp on the . at machine had an addressing scheme that allowed one to reference the “address” and “decrement” parts of a memory location. car stands for “Contents of Address part of Register” and cdr (pronounced “could-er”) stands for “Contents of Decrement part of Register.” 115 (define x (cons 1 2)) (define y (cons 3 4)) (define z (cons x y)) (car (car z)) 1 (car (cdr z)) 3 In Section 2.2 we will see how this ability to combine pairs means that pairs can be used as general-purpose building blocks to create all sorts of complex data structures. e single compound-data primitive pair, implemented by the procedures cons, car, and cdr, is the only glue we need. Data objects constructed from pairs are called list-structured data. Representing rational numbers Pairs oﬀer a natural way to complete the rational-number system. Sim- ply represent a rational number as a pair of two integers: a numerator and a denominator. en make-rat, numer, and denom are readily im- plemented as follows:3 3 Another way to deﬁne the selectors and constructor is (define make-rat cons) (define numer car) (define denom cdr) e ﬁrst deﬁnition associates the name make-rat with the value of the expression cons, which is the primitive procedure that constructs pairs. us make-rat and cons are names for the same primitive constructor. Deﬁning selectors and constructors in this way is eﬃcient: Instead of make-rat call- ing cons, make-rat is cons, so there is only one procedure called, not two, when make- rat is called. On the other hand, doing this defeats debugging aids that trace procedure calls or put breakpoints on procedure calls: You may want to watch make-rat being called, but you certainly don’t want to watch every call to cons. We have chosen not to use this style of deﬁnition in this book. 116 (define (make-rat n d) (cons n d)) (define (numer x) (car x)) (define (denom x) (cdr x)) Also, in order to display the results of our computations, we can print rational numbers by printing the numerator, a slash, and the denomi- nator:4 (define (print-rat x) (newline) (display (numer x)) (display "/") (display (denom x))) Now we can try our rational-number procedures: (define one-half (make-rat 1 2)) (print-rat one-half) 1/2 (define one-third (make-rat 1 3)) (print-rat (add-rat one-half one-third)) 5/6 (print-rat (mul-rat one-half one-third)) 1/6 (print-rat (add-rat one-third one-third)) 6/9 As the ﬁnal example shows, our rational-number implementation does not reduce rational numbers to lowest terms. We can remedy this by changing make-rat. If we have a gcd procedure like the one in Section 1.2.5 that produces the greatest common divisor of two integers, we can 4 display is the Scheme primitive for printing data. e Scheme primitive newline starts a new line for printing. Neither of these procedures returns a useful value, so in the uses of print-rat below, we show only what print-rat prints, not what the interpreter prints as the value returned by print-rat. 117 use gcd to reduce the numerator and the denominator to lowest terms before constructing the pair: (define (make-rat n d) (let ((g (gcd n d))) (cons (/ n g) (/ d g)))) Now we have (print-rat (add-rat one-third one-third)) 2/3 as desired. is modiﬁcation was accomplished by changing the con- structor make-rat without changing any of the procedures (such as add-rat and mul-rat) that implement the actual operations. Exercise 2.1: Deﬁne a beer version of make-rat that han- dles both positive and negative arguments. make-rat should normalize the sign so that if the rational number is positive, both the numerator and denominator are positive, and if the rational number is negative, only the numerator is neg- ative. 2.1.2 Abstraction Barriers Before continuing with more examples of compound data and data ab- straction, let us consider some of the issues raised by the rational-number example. We deﬁned the rational-number operations in terms of a con- structor make-rat and selectors numer and denom. In general, the under- lying idea of data abstraction is to identify for each type of data object a basic set of operations in terms of which all manipulations of data objects of that type will be expressed, and then to use only those oper- ations in manipulating the data. 118 Programs that use rational numbers Rational numbers in problem domain add-rat sub-rat ... Rational numbers as numerators and denominators make-rat numer denom Rational numbers as pairs cons car cdr However pairs are implemented Figure 2.1: Data-abstraction barriers in the rational- number package. We can envision the structure of the rational-number system as shown in Figure 2.1. e horizontal lines represent abstraction barriers that isolate diﬀerent “levels” of the system. At each level, the barrier separates the programs (above) that use the data abstraction from the programs (below) that implement the data abstraction. Programs that use rational numbers manipulate them solely in terms of the proce- dures supplied “for public use” by the rational-number package: add- rat, sub-rat, mul-rat, div-rat, and equal-rat?. ese, in turn, are implemented solely in terms of the constructor and selectors make-rat, numer, and denom, which themselves are implemented in terms of pairs. e details of how pairs are implemented are irrelevant to the rest of the rational-number package so long as pairs can be manipulated by the use of cons, car, and cdr. In eﬀect, procedures at each level are the 119 interfaces that deﬁne the abstraction barriers and connect the diﬀerent levels. is simple idea has many advantages. One advantage is that it makes programs much easier to maintain and to modify. Any complex data structure can be represented in a variety of ways with the prim- itive data structures provided by a programming language. Of course, the choice of representation inﬂuences the programs that operate on it; thus, if the representation were to be changed at some later time, all such programs might have to be modiﬁed accordingly. is task could be time-consuming and expensive in the case of large programs unless the dependence on the representation were to be conﬁned by design to a very few program modules. For example, an alternate way to address the problem of reducing rational numbers to lowest terms is to perform the reduction whenever we access the parts of a rational number, rather than when we construct it. is leads to diﬀerent constructor and selector procedures: (define (make-rat n d) (cons n d)) (define (numer x) (let ((g (gcd (car x) (cdr x)))) (/ (car x) g))) (define (denom x) (let ((g (gcd (car x) (cdr x)))) (/ (cdr x) g))) e diﬀerence between this implementation and the previous one lies in when we compute the gcd. If in our typical use of rational numbers we access the numerators and denominators of the same rational num- bers many times, it would be preferable to compute the gcd when the rational numbers are constructed. If not, we may be beer oﬀ waiting until access time to compute the gcd. In any case, when we change from 120 one representation to the other, the procedures add-rat, sub-rat, and so on do not have to be modiﬁed at all. Constraining the dependence on the representation to a few in- terface procedures helps us design programs as well as modify them, because it allows us to maintain the ﬂexibility to consider alternate implementations. To continue with our simple example, suppose we are designing a rational-number package and we can’t decide initially whether to perform the gcd at construction time or at selection time. e data-abstraction methodology gives us a way to defer that decision without losing the ability to make progress on the rest of the system. Exercise 2.2: Consider the problem of representing line segments in a plane. Each segment is represented as a pair of points: a starting point and an ending point. Deﬁne a constructor make-segment and selectors start-segment and end-segment that deﬁne the representation of segments in terms of points. Furthermore, a point can be represented as a pair of numbers: the x coordinate and the y coordi- nate. Accordingly, specify a constructor make-point and selectors x-point and y-point that deﬁne this representa- tion. Finally, using your selectors and constructors, deﬁne a procedure midpoint-segment that takes a line segment as argument and returns its midpoint (the point whose coor- dinates are the average of the coordinates of the endpoints). To try your procedures, you’ll need a way to print points: (define (print-point p) (newline) (display "(") (display (x-point p)) (display ",") 121 (display (y-point p)) (display ")")) Exercise 2.3: Implement a representation for rectangles in a plane. (Hint: You may want to make use of Exercise 2.2.) In terms of your constructors and selectors, create procedures that compute the perimeter and the area of a given rectan- gle. Now implement a diﬀerent representation for rectan- gles. Can you design your system with suitable abstraction barriers, so that the same perimeter and area procedures will work using either representation? 2.1.3 What Is Meant by Data? We began the rational-number implementation in Section 2.1.1 by im- plementing the rational-number operations add-rat, sub-rat, and so on in terms of three unspeciﬁed procedures: make-rat, numer, and denom. At that point, we could think of the operations as being deﬁned in terms of data objects—numerators, denominators, and rational numbers—whose behavior was speciﬁed by the laer three procedures. But exactly what is meant by data? It is not enough to say “whatever is implemented by the given selectors and constructors.” Clearly, not every arbitrary set of three procedures can serve as an appropriate basis for the rational-number implementation. We need to guarantee that, if we construct a rational number x from a pair of integers n and d, then extracting the numer and the denom of x and dividing them should yield the same result as dividing n by d. In other words, make-rat, numer, and denom must satisfy the condition that, for any integer n and any 122 non-zero integer d, if x is (make-rat n d), then (numer x) n = . (denom x) d In fact, this is the only condition make-rat, numer, and denom must fulﬁll in order to form a suitable basis for a rational-number representation. In general, we can think of data as deﬁned by some collection of se- lectors and constructors, together with speciﬁed conditions that these procedures must fulﬁll in order to be a valid representation.5 is point of view can serve to deﬁne not only “high-level” data ob- jects, such as rational numbers, but lower-level objects as well. Consider the notion of a pair, which we used in order to deﬁne our rational num- bers. We never actually said what a pair was, only that the language supplied procedures cons, car, and cdr for operating on pairs. But the only thing we need to know about these three operations is that if we glue two objects together using cons we can retrieve the objects using car and cdr. at is, the operations satisfy the condition that, for any objects x and y, if z is (cons x y) then (car z) is x and (cdr z) is y. 5 Surprisingly, this idea is very diﬃcult to formulate rigorously. ere are two ap- proaches to giving such a formulation. One, pioneered by C. A. R. Hoare (1972), is known as the method of abstract models. It formalizes the “procedures plus conditions” speciﬁcation as outlined in the rational-number example above. Note that the condi- tion on the rational-number representation was stated in terms of facts about integers (equality and division). In general, abstract models deﬁne new kinds of data objects in terms of previously deﬁned types of data objects. Assertions about data objects can therefore be checked by reducing them to assertions about previously deﬁned data ob- jects. Another approach, introduced by Zilles at , by Goguen, atcher, Wagner, and Wright at (see atcher et al. 1978), and by Guag at Toronto (see Guag 1977), is called algebraic speciﬁcation. It regards the “procedures” as elements of an abstract alge- braic system whose behavior is speciﬁed by axioms that correspond to our “conditions,” and uses the techniques of abstract algebra to check assertions about data objects. Both methods are surveyed in the paper by Liskov and Zilles (1975). 123 Indeed, we mentioned that these three procedures are included as prim- itives in our language. However, any triple of procedures that satisﬁes the above condition can be used as the basis for implementing pairs. is point is illustrated strikingly by the fact that we could implement cons, car, and cdr without using any data structures at all but only using procedures. Here are the deﬁnitions: (define (cons x y) (define (dispatch m) (cond ((= m 0) x) ((= m 1) y) (else (error "Argument not 0 or 1: CONS" m)))) dispatch) (define (car z) (z 0)) (define (cdr z) (z 1)) is use of procedures corresponds to nothing like our intuitive notion of what data should be. Nevertheless, all we need to do to show that this is a valid way to represent pairs is to verify that these procedures satisfy the condition given above. e subtle point to notice is that the value returned by (cons x y) is a procedure—namely the internally deﬁned procedure dispatch, which takes one argument and returns either x or y depending on whether the argument is 0 or 1. Correspondingly, (car z) is deﬁned to apply z to 0. Hence, if z is the procedure formed by (cons x y), then z applied to 0 will yield x. us, we have shown that (car (cons x y)) yields x, as desired. Similarly, (cdr (cons x y)) applies the procedure returned by (cons x y) to 1, which returns y. erefore, this procedural implemen- tation of pairs is a valid implementation, and if we access pairs using only cons, car, and cdr we cannot distinguish this implementation from one that uses “real” data structures. e point of exhibiting the procedural representation of pairs is not 124 that our language works this way (Scheme, and Lisp systems in general, implement pairs directly, for eﬃciency reasons) but that it could work this way. e procedural representation, although obscure, is a perfectly adequate way to represent pairs, since it fulﬁlls the only conditions that pairs need to fulﬁll. is example also demonstrates that the ability to manipulate procedures as objects automatically provides the ability to represent compound data. is may seem a curiosity now, but procedu- ral representations of data will play a central role in our programming repertoire. is style of programming is oen called message passing, and we will be using it as a basic tool in Chapter 3 when we address the issues of modeling and simulation. Exercise 2.4: Here is an alternative procedural representa- tion of pairs. For this representation, verify that (car (cons x y)) yields x for any objects x and y. (define (cons x y) (lambda (m) (m x y))) (define (car z) (z (lambda (p q) p))) What is the corresponding deﬁnition of cdr? (Hint: To ver- ify that this works, make use of the substitution model of Section 1.1.5.) Exercise 2.5: Show that we can represent pairs of nonneg- ative integers using only numbers and arithmetic opera- tions if we represent the pair a and b as the integer that is the product 2a 3b . Give the corresponding deﬁnitions of the procedures cons, car, and cdr. 125 Exercise 2.6: In case representing pairs as procedures wasn’t mind-boggling enough, consider that, in a language that can manipulate procedures, we can get by without numbers (at least insofar as nonnegative integers are concerned) by implementing 0 and the operation of adding 1 as (define zero (lambda (f) (lambda (x) x))) (define (add-1 n) (lambda (f) (lambda (x) (f ((n f) x))))) is representation is known as Church numerals, aer its inventor, Alonzo Church, the logician who invented the λ- calculus. Deﬁne one and two directly (not in terms of zero and add- 1). (Hint: Use substitution to evaluate (add-1 zero)). Give a direct deﬁnition of the addition procedure + (not in terms of repeated application of add-1). 2.1.4 Extended Exercise: Interval Arithmetic Alyssa P. Hacker is designing a system to help people solve engineer- ing problems. One feature she wants to provide in her system is the ability to manipulate inexact quantities (such as measured parameters of physical devices) with known precision, so that when computations are done with such approximate quantities the results will be numbers of known precision. Electrical engineers will be using Alyssa’s system to compute elec- trical quantities. It is sometimes necessary for them to compute the value of a parallel equivalent resistance R p of two resistors R 1 , R 2 using the formula 1 Rp = . 1/R 1 + 1/R 2 126 Resistance values are usually known only up to some tolerance guaran- teed by the manufacturer of the resistor. For example, if you buy a resis- tor labeled “6.8 ohms with 10% tolerance” you can only be sure that the resistor has a resistance between 6.8 − 0.68 = 6.12 and 6.8 + 0.68 = 7.48 ohms. us, if you have a 6.8-ohm 10% resistor in parallel with a 4.7-ohm 5% resistor, the resistance of the combination can range from about 2.58 ohms (if the two resistors are at the lower bounds) to about 2.97 ohms (if the two resistors are at the upper bounds). Alyssa’s idea is to implement “interval arithmetic” as a set of arith- metic operations for combining “intervals” (objects that represent the range of possible values of an inexact quantity). e result of adding, subtracting, multiplying, or dividing two intervals is itself an interval, representing the range of the result. Alyssa postulates the existence of an abstract object called an “in- terval” that has two endpoints: a lower bound and an upper bound. She also presumes that, given the endpoints of an interval, she can con- struct the interval using the data constructor make-interval. Alyssa ﬁrst writes a procedure for adding two intervals. She reasons that the minimum value the sum could be is the sum of the two lower bounds and the maximum value it could be is the sum of the two upper bounds: (define (add-interval x y) (make-interval (+ (lower-bound x) (lower-bound y)) (+ (upper-bound x) (upper-bound y)))) Alyssa also works out the product of two intervals by ﬁnding the min- imum and the maximum of the products of the bounds and using them as the bounds of the resulting interval. (min and max are primitives that ﬁnd the minimum or maximum of any number of arguments.) (define (mul-interval x y) (let ((p1 (* (lower-bound x) (lower-bound y))) 127 (p2 (* (lower-bound x) (upper-bound y))) (p3 (* (upper-bound x) (lower-bound y))) (p4 (* (upper-bound x) (upper-bound y)))) (make-interval (min p1 p2 p3 p4) (max p1 p2 p3 p4)))) To divide two intervals, Alyssa multiplies the ﬁrst by the reciprocal of the second. Note that the bounds of the reciprocal interval are the re- ciprocal of the upper bound and the reciprocal of the lower bound, in that order. (define (div-interval x y) (mul-interval x (make-interval (/ 1.0 (upper-bound y)) (/ 1.0 (lower-bound y))))) Exercise 2.7: Alyssa’s program is incomplete because she has not speciﬁed the implementation of the interval ab- straction. Here is a deﬁnition of the interval constructor: (define (make-interval a b) (cons a b)) Deﬁne selectors upper-bound and lower-bound to complete the implementation. Exercise 2.8: Using reasoning analogous to Alyssa’s, de- scribe how the diﬀerence of two intervals may be com- puted. Deﬁne a corresponding subtraction procedure, called sub-interval. Exercise 2.9: e width of an interval is half of the diﬀer- ence between its upper and lower bounds. e width is a 128 measure of the uncertainty of the number speciﬁed by the interval. For some arithmetic operations the width of the result of combining two intervals is a function only of the widths of the argument intervals, whereas for others the width of the combination is not a function of the widths of the argument intervals. Show that the width of the sum (or diﬀerence) of two intervals is a function only of the widths of the intervals being added (or subtracted). Give examples to show that this is not true for multiplication or division. Exercise 2.10: Ben Bitdiddle, an expert systems program- mer, looks over Alyssa’s shoulder and comments that it is not clear what it means to divide by an interval that spans zero. Modify Alyssa’s code to check for this condition and to signal an error if it occurs. Exercise 2.11: In passing, Ben also cryptically comments: “By testing the signs of the endpoints of the intervals, it is possible to break mul-interval into nine cases, only one of which requires more than two multiplications.” Rewrite this procedure using Ben’s suggestion. Aer debugging her program, Alyssa shows it to a poten- tial user, who complains that her program solves the wrong problem. He wants a program that can deal with numbers represented as a center value and an additive tolerance; for example, he wants to work with intervals such as 3.5 ± 0.15 rather than [3.35, 3.65]. Alyssa returns to her desk and ﬁxes this problem by supplying an alternate constructor and al- ternate selectors: 129 (define (make-center-width c w) (make-interval (- c w) (+ c w))) (define (center i) (/ (+ (lower-bound i) (upper-bound i)) 2)) (define (width i) (/ (- (upper-bound i) (lower-bound i)) 2)) Unfortunately, most of Alyssa’s users are engineers. Real engineering situations usually involve measurements with only a small uncertainty, measured as the ratio of the width of the interval to the midpoint of the interval. Engineers usually specify percentage tolerances on the parameters of devices, as in the resistor speciﬁcations given earlier. Exercise 2.12: Deﬁne a constructor make-center-percent that takes a center and a percentage tolerance and pro- duces the desired interval. You must also deﬁne a selector percent that produces the percentage tolerance for a given interval. e center selector is the same as the one shown above. Exercise 2.13: Show that under the assumption of small percentage tolerances there is a simple formula for the ap- proximate percentage tolerance of the product of two in- tervals in terms of the tolerances of the factors. You may simplify the problem by assuming that all numbers are pos- itive. Aer considerable work, Alyssa P. Hacker delivers her ﬁn- ished system. Several years later, aer she has forgoen all about it, she gets a frenzied call from an irate user, Lem E. Tweakit. It seems that Lem has noticed that the formula for 130 parallel resistors can be wrien in two algebraically equiv- alent ways: R1 R2 R1 + R2 and 1 . 1/R 1 + 1/R 2 He has wrien the following two programs, each of which computes the parallel-resistors formula diﬀerently: (define (par1 r1 r2) (div-interval (mul-interval r1 r2) (add-interval r1 r2))) (define (par2 r1 r2) (let ((one (make-interval 1 1))) (div-interval one (add-interval (div-interval one r1) (div-interval one r2))))) Lem complains that Alyssa’s program gives diﬀerent an- swers for the two ways of computing. is is a serious com- plaint. Exercise 2.14: Demonstrate that Lem is right. Investigate the behavior of the system on a variety of arithmetic ex- pressions. Make some intervals A and B, and use them in computing the expressions A/A and A/B. You will get the most insight by using intervals whose width is a small per- centage of the center value. Examine the results of the com- putation in center-percent form (see Exercise 2.12). 131 Exercise 2.15: Eva Lu Ator, another user, has also noticed the diﬀerent intervals computed by diﬀerent but algebraically equivalent expressions. She says that a formula to compute with intervals using Alyssa’s system will produce tighter error bounds if it can be wrien in such a form that no vari- able that represents an uncertain number is repeated. us, she says, par2 is a “beer” program for parallel resistances than par1. Is she right? Why? Exercise 2.16: Explain, in general, why equivalent alge- braic expressions may lead to diﬀerent answers. Can you devise an interval-arithmetic package that does not have this shortcoming, or is this task impossible? (Warning: is problem is very diﬃcult.) 2.2 Hierarchical Data and the Closure Property As we have seen, pairs provide a primitive “glue” that we can use to construct compound data objects. Figure 2.2 shows a standard way to visualize a pair—in this case, the pair formed by (cons 1 2). In this representation, which is called box-and-pointer notation, each object is shown as a pointer to a box. e box for a primitive object contains a representation of the object. For example, the box for a number contains a numeral. e box for a pair is actually a double box, the le part con- taining (a pointer to) the car of the pair and the right part containing the cdr. We have already seen that cons can be used to combine not only numbers but pairs as well. (You made use of this fact, or should have, in doing Exercise 2.2 and Exercise 2.3.) As a consequence, pairs pro- vide a universal building block from which we can construct all sorts of 132 2 1 Figure 2.2: Box-and-pointer representation of (cons 1 2). 4 3 4 1 2 1 2 3 (cons (cons 1 2) (cons (cons 1 (cons 3 4)) (cons 2 3)) 4) Figure 2.3: Two ways to combine 1, 2, 3, and 4 using pairs. data structures. Figure 2.3 shows two ways to use pairs to combine the numbers 1, 2, 3, and 4. e ability to create pairs whose elements are pairs is the essence of list structure’s importance as a representational tool. We refer to this ability as the closure property of cons. In general, an operation for com- bining data objects satisﬁes the closure property if the results of com- bining things with that operation can themselves be combined using the same operation.6 Closure is the key to power in any means of combina- 6 e use of the word “closure” here comes from abstract algebra, where a set of 133 tion because it permits us to create hierarchical structures—structures made up of parts, which themselves are made up of parts, and so on. From the outset of Chapter 1, we’ve made essential use of closure in dealing with procedures, because all but the very simplest programs rely on the fact that the elements of a combination can themselves be combinations. In this section, we take up the consequences of closure for compound data. We describe some conventional techniques for us- ing pairs to represent sequences and trees, and we exhibit a graphics language that illustrates closure in a vivid way.7 2.2.1 Representing Sequences One of the useful structures we can build with pairs is a sequence—an ordered collection of data objects. ere are, of course, many ways to elements is said to be closed under an operation if applying the operation to elements in the set produces an element that is again an element of the set. e Lisp community also (unfortunately) uses the word “closure” to describe a totally unrelated concept: A closure is an implementation technique for representing procedures with free variables. We do not use the word “closure” in this second sense in this book. 7 e notion that a means of combination should satisfy closure is a straightfor- ward idea. Unfortunately, the data combiners provided in many popular programming languages do not satisfy closure, or make closure cumbersome to exploit. In Fortran or Basic, one typically combines data elements by assembling them into arrays—but one cannot form arrays whose elements are themselves arrays. Pascal and C admit structures whose elements are structures. However, this requires that the program- mer manipulate pointers explicitly, and adhere to the restriction that each ﬁeld of a structure can contain only elements of a prespeciﬁed form. Unlike Lisp with its pairs, these languages have no built-in general-purpose glue that makes it easy to manipulate compound data in a uniform way. is limitation lies behind Alan Perlis’s comment in his foreword to this book: “In Pascal the plethora of declarable data structures induces a specialization within functions that inhibits and penalizes casual cooperation. It is beer to have 100 functions operate on one data structure than to have 10 functions operate on 10 data structures.” 134 1 2 3 4 Figure 2.4: e sequence 1, 2, 3, 4 represented as a chain of pairs. represent sequences in terms of pairs. One particularly straightforward representation is illustrated in Figure 2.4, where the sequence 1, 2, 3, 4 is represented as a chain of pairs. e car of each pair is the corresponding item in the chain, and the cdr of the pair is the next pair in the chain. e cdr of the ﬁnal pair signals the end of the sequence by pointing to a distinguished value that is not a pair, represented in box-and-pointer diagrams as a diagonal line and in programs as the value of the variable nil. e entire sequence is constructed by nested cons operations: (cons 1 (cons 2 (cons 3 (cons 4 nil)))) Such a sequence of pairs, formed by nested conses, is called a list, and Scheme provides a primitive called list to help in constructing lists.8 e above sequence could be produced by (list 1 2 3 4). In general, (list ⟨a1 ⟩ ⟨a2 ⟩ . . . ⟨an ⟩) 8 In this book, we use list to mean a chain of pairs terminated by the end-of-list marker. In contrast, the term list structure refers to any data structure made out of pairs, not just to lists. 135 is equivalent to (cons ⟨a1 ⟩ (cons ⟨a2 ⟩ (cons ... (cons ⟨an ⟩ nil). . .))) Lisp systems conventionally print lists by printing the sequence of el- ements, enclosed in parentheses. us, the data object in Figure 2.4 is printed as (1 2 3 4): (define one-through-four (list 1 2 3 4)) one-through-four (1 2 3 4) Be careful not to confuse the expression (list 1 2 3 4) with the list (1 2 3 4), which is the result obtained when the expression is evalu- ated. Aempting to evaluate the expression (1 2 3 4) will signal an error when the interpreter tries to apply the procedure 1 to arguments 2, 3, and 4. We can think of car as selecting the ﬁrst item in the list, and of cdr as selecting the sublist consisting of all but the ﬁrst item. Nested applications of car and cdr can be used to extract the second, third, and subsequent items in the list.9 e constructor cons makes a list like the original one, but with an additional item at the beginning. 9 Sincenested applications of car and cdr are cumbersome to write, Lisp dialects provide abbreviations for them—for instance, (cadr ⟨arg⟩) = (car (cdr ⟨arg⟩)) e names of all such procedures start with c and end with r. Each a between them stands for a car operation and each d for a cdr operation, to be applied in the same order in which they appear in the name. e names car and cdr persist because simple combinations like cadr are pronounceable. 136 (car one-through-four) 1 (cdr one-through-four) (2 3 4) (car (cdr one-through-four)) 2 (cons 10 one-through-four) (10 1 2 3 4) (cons 5 one-through-four) (5 1 2 3 4) e value of nil, used to terminate the chain of pairs, can be thought of as a sequence of no elements, the empty list. e word nil is a contraction of the Latin word nihil, which means “nothing.”10 List operations e use of pairs to represent sequences of elements as lists is accompa- nied by conventional programming techniques for manipulating lists by successively “cdring down” the lists. For example, the procedure list- ref takes as arguments a list and a number n and returns the n th item of the list. It is customary to number the elements of the list beginning with 0. e method for computing list-ref is the following: 10 It’s remarkable how much energy in the standardization of Lisp dialects has been dissipated in arguments that are literally over nothing: Should nil be an ordinary name? Should the value of nil be a symbol? Should it be a list? Should it be a pair? In Scheme, nil is an ordinary name, which we use in this section as a variable whose value is the end-of-list marker (just as true is an ordinary variable that has a true value). Other dialects of Lisp, including Common Lisp, treat nil as a special symbol. e au- thors of this book, who have endured too many language standardization brawls, would like to avoid the entire issue. Once we have introduced quotation in Section 2.3, we will denote the empty list as '() and dispense with the variable nil entirely. 137 • For n = 0, list-ref should return the car of the list. • Otherwise, list-ref should return the (n − 1)-st item of the cdr of the list. (define (list-ref items n) (if (= n 0) (car items) (list-ref (cdr items) (- n 1)))) (define squares (list 1 4 9 16 25)) (list-ref squares 3) 16 Oen we cdr down the whole list. To aid in this, Scheme includes a primitive predicate null?, which tests whether its argument is the empty list. e procedure length, which returns the number of items in a list, illustrates this typical paern of use: (define (length items) (if (null? items) 0 (+ 1 (length (cdr items))))) (define odds (list 1 3 5 7)) (length odds) 4 e length procedure implements a simple recursive plan. e reduc- tion step is: • e length of any list is 1 plus the length of the cdr of the list. is is applied successively until we reach the base case: • e length of the empty list is 0. 138 We could also compute length in an iterative style: (define (length items) (define (length-iter a count) (if (null? a) count (length-iter (cdr a) (+ 1 count)))) (length-iter items 0)) Another conventional programming technique is to “cons up” an an- swer list while cdring down a list, as in the procedure append, which takes two lists as arguments and combines their elements to make a new list: (append squares odds) (1 4 9 16 25 1 3 5 7) (append odds squares) (1 3 5 7 1 4 9 16 25) append is also implemented using a recursive plan. To append lists list1 and list2, do the following: • If list1 is the empty list, then the result is just list2. • Otherwise, append the cdr of list1 and list2, and cons the car of list1 onto the result: (define (append list1 list2) (if (null? list1) list2 (cons (car list1) (append (cdr list1) list2)))) Exercise 2.17: Deﬁne a procedure last-pair that returns the list that contains only the last element of a given (nonempty) list: 139 (last-pair (list 23 72 149 34)) (34) Exercise 2.18: Deﬁne a procedure reverse that takes a list as argument and returns a list of the same elements in re- verse order: (reverse (list 1 4 9 16 25)) (25 16 9 4 1) Exercise 2.19: Consider the change-counting program of Section 1.2.2. It would be nice to be able to easily change the currency used by the program, so that we could compute the number of ways to change a British pound, for example. As the program is wrien, the knowledge of the currency is distributed partly into the procedure first-denomination and partly into the procedure count-change (which knows that there are ﬁve kinds of U.S. coins). It would be nicer to be able to supply a list of coins to be used for making change. We want to rewrite the procedure cc so that its second ar- gument is a list of the values of the coins to use rather than an integer specifying which coins to use. We could then have lists that deﬁned each kind of currency: (define us-coins (list 50 25 10 5 1)) (define uk-coins (list 100 50 20 10 5 2 1 0.5)) We could then call cc as follows: (cc 100 us-coins) 292 140 To do this will require changing the program cc somewhat. It will still have the same form, but it will access its second argument diﬀerently, as follows: (define (cc amount coin-values) (cond ((= amount 0) 1) ((or (< amount 0) (no-more? coin-values)) 0) (else (+ (cc amount (except-first-denomination coin-values)) (cc (- amount (first-denomination coin-values)) coin-values))))) Deﬁne the procedures first-denomination, except-first- denomination, and no-more? in terms of primitive oper- ations on list structures. Does the order of the list coin- values aﬀect the answer produced by cc? Why or why not? Exercise 2.20: e procedures +, *, and list take arbitrary numbers of arguments. One way to deﬁne such procedures is to use define with doed-tail notation. In a procedure deﬁnition, a parameter list that has a dot before the last pa- rameter name indicates that, when the procedure is called, the initial parameters (if any) will have as values the initial arguments, as usual, but the ﬁnal parameter’s value will be a list of any remaining arguments. For instance, given the deﬁnition (define (f x y . z) ⟨body⟩) 141 the procedure f can be called with two or more arguments. If we evaluate (f 1 2 3 4 5 6) then in the body of f, x will be 1, y will be 2, and z will be the list (3 4 5 6). Given the deﬁnition (define (g . w) ⟨body⟩) the procedure g can be called with zero or more arguments. If we evaluate (g 1 2 3 4 5 6) then in the body of g, w will be the list (1 2 3 4 5 6).11 Use this notation to write a procedure same-parity that takes one or more integers and returns a list of all the ar- guments that have the same even-odd parity as the ﬁrst argument. For example, (same-parity 1 2 3 4 5 6 7) (1 3 5 7) (same-parity 2 3 4 5 6 7) (2 4 6) 11 To deﬁne f and g using lambda we would write (define f (lambda (x y . z) ⟨body⟩)) (define g (lambda w ⟨body⟩)) 142 Mapping over lists One extremely useful operation is to apply some transformation to each element in a list and generate the list of results. For instance, the follow- ing procedure scales each number in a list by a given factor: (define (scale-list items factor) (if (null? items) nil (cons (* (car items) factor) (scale-list (cdr items) factor)))) (scale-list (list 1 2 3 4 5) 10) (10 20 30 40 50) We can abstract this general idea and capture it as a common paern expressed as a higher-order procedure, just as in Section 1.3. e higher- order procedure here is called map. map takes as arguments a procedure of one argument and a list, and returns a list of the results produced by applying the procedure to each element in the list:12 (define (map proc items) (if (null? items) 12 Scheme standardly provides a map procedure that is more general than the one described here. is more general map takes a procedure of n arguments, together with n lists, and applies the procedure to all the ﬁrst elements of the lists, all the second elements of the lists, and so on, returning a list of the results. For example: (map + (list 1 2 3) (list 40 50 60) (list 700 800 900)) (741 852 963) (map (lambda (x y) (+ x (* 2 y))) (list 1 2 3) (list 4 5 6)) (9 12 15) 143 nil (cons (proc (car items)) (map proc (cdr items))))) (map abs (list -10 2.5 -11.6 17)) (10 2.5 11.6 17) (map (lambda (x) (* x x)) (list 1 2 3 4)) (1 4 9 16) Now we can give a new deﬁnition of scale-list in terms of map: (define (scale-list items factor) (map (lambda (x) (* x factor)) items)) map is an important construct, not only because it captures a common paern, but because it establishes a higher level of abstraction in dealing with lists. In the original deﬁnition of scale-list, the recursive struc- ture of the program draws aention to the element-by-element process- ing of the list. Deﬁning scale-list in terms of map suppresses that level of detail and emphasizes that scaling transforms a list of elements to a list of results. e diﬀerence between the two deﬁnitions is not that the computer is performing a diﬀerent process (it isn’t) but that we think about the process diﬀerently. In eﬀect, map helps establish an abstrac- tion barrier that isolates the implementation of procedures that trans- form lists from the details of how the elements of the list are extracted and combined. Like the barriers shown in Figure 2.1, this abstraction gives us the ﬂexibility to change the low-level details of how sequences are implemented, while preserving the conceptual framework of oper- ations that transform sequences to sequences. Section 2.2.3 expands on this use of sequences as a framework for organizing programs. Exercise 2.21: e procedure square-list takes a list of numbers as argument and returns a list of the squares of 144 those numbers. (square-list (list 1 2 3 4)) (1 4 9 16) Here are two diﬀerent deﬁnitions of square-list. Com- plete both of them by ﬁlling in the missing expressions: (define (square-list items) (if (null? items) nil (cons ⟨??⟩ ⟨??⟩))) (define (square-list items) (map ⟨??⟩ ⟨??⟩)) Exercise 2.22: Louis Reasoner tries to rewrite the ﬁrst square- list procedure of Exercise 2.21 so that it evolves an itera- tive process: (define (square-list items) (define (iter things answer) (if (null? things) answer (iter (cdr things) (cons (square (car things)) answer)))) (iter items nil)) Unfortunately, deﬁning square-list this way produces the answer list in the reverse order of the one desired. Why? Louis then tries to ﬁx his bug by interchanging the argu- ments to cons: (define (square-list items) 145 (define (iter things answer) (if (null? things) answer (iter (cdr things) (cons answer (square (car things)))))) (iter items nil)) is doesn’t work either. Explain. Exercise 2.23: e procedure for-each is similar to map. It takes as arguments a procedure and a list of elements. How- ever, rather than forming a list of the results, for-each just applies the procedure to each of the elements in turn, from le to right. e values returned by applying the procedure to the elements are not used at all—for-each is used with procedures that perform an action, such as printing. For ex- ample, (for-each (lambda (x) (newline) (display x)) (list 57 321 88)) 57 321 88 e value returned by the call to for-each (not illustrated above) can be something arbitrary, such as true. Give an implementation of for-each. 146 (3 4) ((1 2) 3 4) (1 2) 3 4 1 2 Figure 2.5: Structure formed by (cons (list 1 2) (list 3 4)). 2.2.2 Hierarchical Structures e representation of sequences in terms of lists generalizes naturally to represent sequences whose elements may themselves be sequences. For example, we can regard the object ((1 2) 3 4) constructed by (cons (list 1 2) (list 3 4)) as a list of three items, the ﬁrst of which is itself a list, (1 2). Indeed, this is suggested by the form in which the result is printed by the interpreter. Figure 2.5 shows the representation of this structure in terms of pairs. Another way to think of sequences whose elements are sequences is as trees. e elements of the sequence are the branches of the tree, and elements that are themselves sequences are subtrees. Figure 2.6 shows the structure in Figure 2.5 viewed as a tree. Recursion is a natural tool for dealing with tree structures, since we can oen reduce operations on trees to operations on their branches, which reduce in turn to operations on the branches of the branches, and so on, until we reach the leaves of the tree. As an example, compare the 147 ((1 2) 3 4) (1 2) 3 4 1 2 Figure 2.6: e list structure in Figure 2.5 viewed as a tree. length procedure of Section 2.2.1 with the count-leaves procedure, which returns the total number of leaves of a tree: (define x (cons (list 1 2) (list 3 4))) (length x) 3 (count-leaves x) 4 (list x x) (((1 2) 3 4) ((1 2) 3 4)) (length (list x x)) 2 (count-leaves (list x x)) 8 To implement count-leaves, recall the recursive plan for computing length: • length of a list x is 1 plus length of the cdr of x. • length of the empty list is 0. count-leaves is similar. e value for the empty list is the same: • count-leaves of the empty list is 0. 148 But in the reduction step, where we strip oﬀ the car of the list, we must take into account that the car may itself be a tree whose leaves we need to count. us, the appropriate reduction step is • count-leaves of a tree x is count-leaves of the car of x plus count-leaves of the cdr of x. Finally, by taking cars we reach actual leaves, so we need another base case: • count-leaves of a leaf is 1. To aid in writing recursive procedures on trees, Scheme provides the primitive predicate pair?, which tests whether its argument is a pair. Here is the complete procedure:13 (define (count-leaves x) (cond ((null? x) 0) ((not (pair? x)) 1) (else (+ (count-leaves (car x)) (count-leaves (cdr x)))))) Exercise 2.24: Suppose we evaluate the expression (list 1 (list 2 (list 3 4))). Give the result printed by the interpreter, the corresponding box-and-pointer structure, and the interpretation of this as a tree (as in Figure 2.6). Exercise 2.25: Give combinations of cars and cdrs that will pick 7 from each of the following lists: 13 e order of the ﬁrst two clauses in the cond maers, since the empty list satisﬁes null? and also is not a pair. 149 (1 3 (5 7) 9) ((7)) (1 (2 (3 (4 (5 (6 7)))))) Exercise 2.26: Suppose we deﬁne x and y to be two lists: (define x (list 1 2 3)) (define y (list 4 5 6)) What result is printed by the interpreter in response to eval- uating each of the following expressions: (append x y) (cons x y) (list x y) Exercise 2.27: Modify your reverse procedure of Exercise 2.18 to produce a deep-reverse procedure that takes a list as argument and returns as its value the list with its ele- ments reversed and with all sublists deep-reversed as well. For example, (define x (list (list 1 2) (list 3 4))) x ((1 2) (3 4)) (reverse x) ((3 4) (1 2)) (deep-reverse x) ((4 3) (2 1)) Exercise 2.28: Write a procedure fringe that takes as argu- ment a tree (represented as a list) and returns a list whose elements are all the leaves of the tree arranged in le-to- right order. For example, 150 (define x (list (list 1 2) (list 3 4))) (fringe x) (1 2 3 4) (fringe (list x x)) (1 2 3 4 1 2 3 4) Exercise 2.29: A binary mobile consists of two branches, a le branch and a right branch. Each branch is a rod of a certain length, from which hangs either a weight or an- other binary mobile. We can represent a binary mobile us- ing compound data by constructing it from two branches (for example, using list): (define (make-mobile left right) (list left right)) A branch is constructed from a length (which must be a number) together with a structure, which may be either a number (representing a simple weight) or another mobile: (define (make-branch length structure) (list length structure)) a. Write the corresponding selectors left-branch and right-branch, which return the branches of a mobile, and branch-length and branch-structure, which re- turn the components of a branch. b. Using your selectors, deﬁne a procedure total-weight that returns the total weight of a mobile. c. A mobile is said to be balanced if the torque applied by its top-le branch is equal to that applied by its top- 151 right branch (that is, if the length of the le rod mul- tiplied by the weight hanging from that rod is equal to the corresponding product for the right side) and if each of the submobiles hanging oﬀ its branches is bal- anced. Design a predicate that tests whether a binary mobile is balanced. d. Suppose we change the representation of mobiles so that the constructors are (define (make-mobile left right) (cons left right)) (define (make-branch length structure) (cons length structure)) How much do you need to change your programs to convert to the new representation? Mapping over trees Just as map is a powerful abstraction for dealing with sequences, map together with recursion is a powerful abstraction for dealing with trees. For instance, the scale-tree procedure, analogous to scale-list of Section 2.2.1, takes as arguments a numeric factor and a tree whose leaves are numbers. It returns a tree of the same shape, where each number is multiplied by the factor. e recursive plan for scale-tree is similar to the one for count-leaves: (define (scale-tree tree factor) (cond ((null? tree) nil) ((not (pair? tree)) (* tree factor)) (else (cons (scale-tree (car tree) factor) (scale-tree (cdr tree) factor))))) (scale-tree (list 1 (list 2 (list 3 4) 5) (list 6 7)) 10) (10 (20 (30 40) 50) (60 70)) 152 Another way to implement scale-tree is to regard the tree as a se- quence of sub-trees and use map. We map over the sequence, scaling each sub-tree in turn, and return the list of results. In the base case, where the tree is a leaf, we simply multiply by the factor: (define (scale-tree tree factor) (map (lambda (sub-tree) (if (pair? sub-tree) (scale-tree sub-tree factor) (* sub-tree factor))) tree)) Many tree operations can be implemented by similar combinations of sequence operations and recursion. Exercise 2.30: Deﬁne a procedure square-tree analogous to the square-list procedure of Exercise 2.21. at is, square- tree should behave as follows: (square-tree (list 1 (list 2 (list 3 4) 5) (list 6 7))) (1 (4 (9 16) 25) (36 49)) Deﬁne square-tree both directly (i.e., without using any higher-order procedures) and also by using map and recur- sion. Exercise 2.31: Abstract your answer to Exercise 2.30 to produce a procedure tree-map with the property that square- tree could be deﬁned as (define (square-tree tree) (tree-map square tree)) 153 Exercise 2.32: We can represent a set as a list of distinct elements, and we can represent the set of all subsets of the set as a list of lists. For example, if the set is (1 2 3), then the set of all subsets is (() (3) (2) (2 3) (1) (1 3) (1 2) (1 2 3)). Complete the following deﬁnition of a procedure that generates the set of subsets of a set and give a clear explanation of why it works: (define (subsets s) (if (null? s) (list nil) (let ((rest (subsets (cdr s)))) (append rest (map ⟨??⟩ rest))))) 2.2.3 Sequences as Conventional Interfaces In working with compound data, we’ve stressed how data abstraction permits us to design programs without becoming enmeshed in the de- tails of data representations, and how abstraction preserves for us the ﬂexibility to experiment with alternative representations. In this sec- tion, we introduce another powerful design principle for working with data structures—the use of conventional interfaces. In Section 1.3 we saw how program abstractions, implemented as higher-order procedures, can capture common paerns in programs that deal with numerical data. Our ability to formulate analogous oper- ations for working with compound data depends crucially on the style in which we manipulate our data structures. Consider, for example, the following procedure, analogous to the count-leaves procedure of Sec- tion 2.2.2, which takes a tree as argument and computes the sum of the squares of the leaves that are odd: 154 (define (sum-odd-squares tree) (cond ((null? tree) 0) ((not (pair? tree)) (if (odd? tree) (square tree) 0)) (else (+ (sum-odd-squares (car tree)) (sum-odd-squares (cdr tree)))))) On the surface, this procedure is very diﬀerent from the following one, which constructs a list of all the even Fibonacci numbers Fib(k), where k is less than or equal to a given integer n: (define (even-fibs n) (define (next k) (if (> k n) nil (let ((f (fib k))) (if (even? f) (cons f (next (+ k 1))) (next (+ k 1)))))) (next 0)) Despite the fact that these two procedures are structurally very diﬀer- ent, a more abstract description of the two computations reveals a great deal of similarity. e ﬁrst program • enumerates the leaves of a tree; • ﬁlters them, selecting the odd ones; • squares each of the selected ones; and • accumulates the results using +, starting with 0. e second program 155 enumerate: filter: map: accumulate: tree leaves odd? square +, 0 enumerate: map: filter: accumulate: integers fib even? cons, () Figure 2.7: e signal-ﬂow plans for the procedures sum- odd-squares (top) and even-fibs (boom) reveal the com- monality between the two programs. • enumerates the integers from 0 to n; • computes the Fibonacci number for each integer; • ﬁlters them, selecting the even ones; and • accumulates the results using cons, starting with the empty list. A signal-processing engineer would ﬁnd it natural to conceptualize these processes in terms of signals ﬂowing through a cascade of stages, each of which implements part of the program plan, as shown in Figure 2.7. In sum-odd-squares, we begin with an enumerator, which generates a “signal” consisting of the leaves of a given tree. is signal is passed through a ﬁlter, which eliminates all but the odd elements. e result- ing signal is in turn passed through a map, which is a “transducer” that applies the square procedure to each element. e output of the map is then fed to an accumulator, which combines the elements using +, starting from an initial 0. e plan for even-fibs is analogous. Unfortunately, the two procedure deﬁnitions above fail to exhibit this signal-ﬂow structure. For instance, if we examine the sum-odd- 156 squares procedure, we ﬁnd that the enumeration is implemented partly by the null? and pair? tests and partly by the tree-recursive structure of the procedure. Similarly, the accumulation is found partly in the tests and partly in the addition used in the recursion. In general, there are no distinct parts of either procedure that correspond to the elements in the signal-ﬂow description. Our two procedures decompose the computa- tions in a diﬀerent way, spreading the enumeration over the program and mingling it with the map, the ﬁlter, and the accumulation. If we could organize our programs to make the signal-ﬂow structure manifest in the procedures we write, this would increase the conceptual clarity of the resulting code. Sequence Operations e key to organizing programs so as to more clearly reﬂect the signal- ﬂow structure is to concentrate on the “signals” that ﬂow from one stage in the process to the next. If we represent these signals as lists, then we can use list operations to implement the processing at each of the stages. For instance, we can implement the mapping stages of the signal-ﬂow diagrams using the map procedure from Section 2.2.1: (map square (list 1 2 3 4 5)) (1 4 9 16 25) Filtering a sequence to select only those elements that satisfy a given predicate is accomplished by (define (filter predicate sequence) (cond ((null? sequence) nil) ((predicate (car sequence)) (cons (car sequence) (filter predicate (cdr sequence)))) (else (filter predicate (cdr sequence))))) 157 For example, (filter odd? (list 1 2 3 4 5)) (1 3 5) Accumulations can be implemented by (define (accumulate op initial sequence) (if (null? sequence) initial (op (car sequence) (accumulate op initial (cdr sequence))))) (accumulate + 0 (list 1 2 3 4 5)) 15 (accumulate * 1 (list 1 2 3 4 5)) 120 (accumulate cons nil (list 1 2 3 4 5)) (1 2 3 4 5) All that remains to implement signal-ﬂow diagrams is to enumerate the sequence of elements to be processed. For even-fibs, we need to gen- erate the sequence of integers in a given range, which we can do as follows: (define (enumerate-interval low high) (if (> low high) nil (cons low (enumerate-interval (+ low 1) high)))) (enumerate-interval 2 7) (2 3 4 5 6 7) To enumerate the leaves of a tree, we can use14 14 isis, in fact, precisely the fringe procedure from Exercise 2.28. Here we’ve re- named it to emphasize that it is part of a family of general sequence-manipulation procedures. 158 (define (enumerate-tree tree) (cond ((null? tree) nil) ((not (pair? tree)) (list tree)) (else (append (enumerate-tree (car tree)) (enumerate-tree (cdr tree)))))) (enumerate-tree (list 1 (list 2 (list 3 4)) 5)) (1 2 3 4 5) Now we can reformulate sum-odd-squares and even-fibs as in the signal-ﬂow diagrams. For sum-odd-squares, we enumerate the sequence of leaves of the tree, ﬁlter this to keep only the odd numbers in the se- quence, square each element, and sum the results: (define (sum-odd-squares tree) (accumulate + 0 (map square (filter odd? (enumerate-tree tree))))) For even-fibs, we enumerate the integers from 0 to n, generate the Fi- bonacci number for each of these integers, ﬁlter the resulting sequence to keep only the even elements, and accumulate the results into a list: (define (even-fibs n) (accumulate cons nil (filter even? (map fib (enumerate-interval 0 n))))) e value of expressing programs as sequence operations is that this helps us make program designs that are modular, that is, designs that are constructed by combining relatively independent pieces. We can en- courage modular design by providing a library of standard components together with a conventional interface for connecting the components in ﬂexible ways. 159 Modular construction is a powerful strategy for controlling com- plexity in engineering design. In real signal-processing applications, for example, designers regularly build systems by cascading elements se- lected from standardized families of ﬁlters and transducers. Similarly, sequence operations provide a library of standard program elements that we can mix and match. For instance, we can reuse pieces from the sum-odd-squares and even-fibs procedures in a program that con- structs a list of the squares of the ﬁrst n + 1 Fibonacci numbers: (define (list-fib-squares n) (accumulate cons nil (map square (map fib (enumerate-interval 0 n))))) (list-fib-squares 10) (0 1 1 4 9 25 64 169 441 1156 3025) We can rearrange the pieces and use them in computing the product of the squares of the odd integers in a sequence: (define (product-of-squares-of-odd-elements sequence) (accumulate * 1 (map square (filter odd? sequence)))) (product-of-squares-of-odd-elements (list 1 2 3 4 5)) 225 We can also formulate conventional data-processing applications in terms of sequence operations. Suppose we have a sequence of personnel records and we want to ﬁnd the salary of the highest-paid programmer. Assume that we have a selector salary that returns the salary of a record, and a predicate programmer? that tests if a record is for a programmer. en we can write (define (salary-of-highest-paid-programmer records) (accumulate max 0 (map salary (filter programmer? records)))) 160 ese examples give just a hint of the vast range of operations that can be expressed as sequence operations.15 Sequences, implemented here as lists, serve as a conventional in- terface that permits us to combine processing modules. Additionally, when we uniformly represent structures as sequences, we have local- ized the data-structure dependencies in our programs to a small number of sequence operations. By changing these, we can experiment with al- ternative representations of sequences, while leaving the overall design of our programs intact. We will exploit this capability in Section 3.5, when we generalize the sequence-processing paradigm to admit inﬁ- nite sequences. Exercise 2.33: Fill in the missing expressions to complete the following deﬁnitions of some basic list-manipulation operations as accumulations: (define (map p sequence) (accumulate (lambda (x y) ⟨??⟩) nil sequence)) (define (append seq1 seq2) (accumulate cons ⟨??⟩ ⟨??⟩)) (define (length sequence) (accumulate ⟨??⟩ 0 sequence)) 15 Richard Waters (1979) developed a program that automatically analyzes traditional Fortran programs, viewing them in terms of maps, ﬁlters, and accumulations. He found that fully 90 percent of the code in the Fortran Scientiﬁc Subroutine Package ﬁts neatly into this paradigm. One of the reasons for the success of Lisp as a programming lan- guage is that lists provide a standard medium for expressing ordered collections so that they can be manipulated using higher-order operations. e programming language APL owes much of its power and appeal to a similar choice. In APL all data are repre- sented as arrays, and there is a universal and convenient set of generic operators for all sorts of array operations. 161 Exercise 2.34: Evaluating a polynomial in x at a given value of x can be formulated as an accumulation. We evaluate the polynomial an x n + an −1x n −1 + · · · + a 1x + a 0 using a well-known algorithm called Horner’s rule, which structures the computation as (. . . (an x + an −1 )x + · · · + a 1 )x + a 0 . In other words, we start with an , multiply by x, add an −1 , multiply by x, and so on, until we reach a 0 .16 Fill in the following template to produce a procedure that evaluates a polynomial using Horner’s rule. Assume that the coeﬃcients of the polynomial are arranged in a sequence, from a 0 through an . (define (horner-eval x coefficient-sequence) (accumulate (lambda (this-coeff higher-terms) ⟨??⟩) 0 coefficient-sequence)) 16 According to Knuth 1981, this rule was formulated by W. G. Horner early in the nineteenth century, but the method was actually used by Newton over a hundred years earlier. Horner’s rule evaluates the polynomial using fewer additions and multipli- cations than does the straightforward method of ﬁrst computing an x n , then adding an −1 x n −1 , and so on. In fact, it is possible to prove that any algorithm for evaluating arbitrary polynomials must use at least as many additions and multiplications as does Horner’s rule, and thus Horner’s rule is an optimal algorithm for polynomial evaluation. is was proved (for the number of additions) by A. M. Ostrowski in a 1954 paper that essentially founded the modern study of optimal algorithms. e analogous statement for multiplications was proved by V. Y. Pan in 1966. e book by Borodin and Munro (1975) provides an overview of these and other results about optimal algorithms. 162 For example, to compute 1+3x +5x 3 +x 5 at x = 2 you would evaluate (horner-eval 2 (list 1 3 0 5 0 1)) Exercise 2.35: Redeﬁne count-leaves from Section 2.2.2 as an accumulation: (define (count-leaves t) (accumulate ⟨??⟩ ⟨??⟩ (map ⟨??⟩ ⟨??⟩))) Exercise 2.36: e procedure accumulate-n is similar to accumulate except that it takes as its third argument a se- quence of sequences, which are all assumed to have the same number of elements. It applies the designated accu- mulation procedure to combine all the ﬁrst elements of the sequences, all the second elements of the sequences, and so on, and returns a sequence of the results. For instance, if s is a sequence containing four sequences, ((1 2 3) (4 5 6) (7 8 9) (10 11 12)), then the value of (accumulate-n + 0 s) should be the sequence (22 26 30). Fill in the missing expressions in the following deﬁnition of accumulate-n: (define (accumulate-n op init seqs) (if (null? (car seqs)) nil (cons (accumulate op init ⟨??⟩) (accumulate-n op init ⟨??⟩)))) Exercise 2.37: Suppose we represent vectors v = (vi ) as sequences of numbers, and matrices m = (mij ) as sequences 163 of vectors (the rows of the matrix). For example, the matrix 1 2 3 4 4 5 6 6 6 7 8 9 is represented as the sequence ((1 2 3 4) (4 5 6 6) (6 7 8 9)). With this representation, we can use sequence operations to concisely express the basic matrix and vector operations. ese operations (which are described in any book on matrix algebra) are the following: returns the sum Σi vi wi ; (dot-product v w) (matrix-*-vector m v) returns the vector t, where ti = Σj mij vj ; (matrix-*-matrix m n) returns the matrix p , where pij = Σk mik n kj ; (transpose m) returns the matrix n , where nij = mji . We can deﬁne the dot product as17 (define (dot-product v w) (accumulate + 0 (map * v w))) Fill in the missing expressions in the following procedures for computing the other matrix operations. (e procedure accumulate-n is deﬁned in Exercise 2.36.) (define (matrix-*-vector m v) (map ⟨??⟩ m)) 17 is deﬁnition uses the extended version of map described in Footnote 12. 164 (define (transpose mat) (accumulate-n ⟨??⟩ ⟨??⟩ mat)) (define (matrix-*-matrix m n) (let ((cols (transpose n))) (map ⟨??⟩ m))) Exercise 2.38: e accumulate procedure is also known as fold-right, because it combines the ﬁrst element of the se- quence with the result of combining all the elements to the right. ere is also a fold-left, which is similar to fold- right, except that it combines elements working in the op- posite direction: (define (fold-left op initial sequence) (define (iter result rest) (if (null? rest) result (iter (op result (car rest)) (cdr rest)))) (iter initial sequence)) What are the values of (fold-right / 1 (list 1 2 3)) (fold-left / 1 (list 1 2 3)) (fold-right list nil (list 1 2 3)) (fold-left list nil (list 1 2 3)) Give a property that op should satisfy to guarantee that fold-right and fold-left will produce the same values for any sequence. 165 Exercise 2.39: Complete the following deﬁnitions of reverse (Exercise 2.18) in terms of fold-right and fold-left from Exercise 2.38: (define (reverse sequence) (fold-right (lambda (x y) ⟨??⟩) nil sequence)) (define (reverse sequence) (fold-left (lambda (x y) ⟨??⟩) nil sequence)) Nested Mappings We can extend the sequence paradigm to include many computations that are commonly expressed using nested loops.18 Consider this prob- lem: Given a positive integer n, ﬁnd all ordered pairs of distinct positive integers i and j, where 1 ≤ j < i ≤ n, such that i + j is prime. For example, if n is 6, then the pairs are the following: i 2 3 4 4 5 6 6 j 1 2 1 3 2 1 5 i+j 3 5 5 7 7 7 11 A natural way to organize this computation is to generate the sequence of all ordered pairs of positive integers less than or equal to n, ﬁlter to select those pairs whose sum is prime, and then, for each pair (i, j) that passes through the ﬁlter, produce the triple (i, j, i + j). Here is a way to generate the sequence of pairs: For each integer i ≤ n, enumerate the integers j < i, and for each such i and j gener- ate the pair (i, j). In terms of sequence operations, we map along the 18 is approach to nested mappings was shown to us by David Turner, whose lan- guages KRC and Miranda provide elegant formalisms for dealing with these constructs. e examples in this section (see also Exercise 2.42) are adapted from Turner 1981. In Section 3.5.3, we’ll see how this approach generalizes to inﬁnite sequences. 166 sequence (enumerate-interval 1 n). For each i in this sequence, we map along the sequence (enumerate-interval 1 (- i 1)). For each j in this laer sequence, we generate the pair (list i j). is gives us a sequence of pairs for each i. Combining all the sequences for all the i (by accumulating with append) produces the required sequence of pairs:19 (accumulate append nil (map (lambda (i) (map (lambda (j) (list i j)) (enumerate-interval 1 (- i 1)))) (enumerate-interval 1 n))) e combination of mapping and accumulating with append is so com- mon in this sort of program that we will isolate it as a separate proce- dure: (define (flatmap proc seq) (accumulate append nil (map proc seq))) Now ﬁlter this sequence of pairs to ﬁnd those whose sum is prime. e ﬁlter predicate is called for each element of the sequence; its argument is a pair and it must extract the integers from the pair. us, the predicate to apply to each element in the sequence is (define (prime-sum? pair) (prime? (+ (car pair) (cadr pair)))) Finally, generate the sequence of results by mapping over the ﬁltered pairs using the following procedure, which constructs a triple consisting of the two elements of the pair along with their sum: 19 We’rerepresenting a pair here as a list of two elements rather than as a Lisp pair. us, the “pair” (i, j) is represented as (list i j), not (cons i j). 167 (define (make-pair-sum pair) (list (car pair) (cadr pair) (+ (car pair) (cadr pair)))) Combining all these steps yields the complete procedure: (define (prime-sum-pairs n) (map make-pair-sum (filter prime-sum? (flatmap (lambda (i) (map (lambda (j) (list i j)) (enumerate-interval 1 (- i 1)))) (enumerate-interval 1 n))))) Nested mappings are also useful for sequences other than those that enumerate intervals. Suppose we wish to generate all the permutations of a set S; that is, all the ways of ordering the items in the set. For in- stance, the permutations of {1, 2, 3} are {1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, and {3, 2, 1}. Here is a plan for generating the permutations of S: For each item x in S, recursively generate the sequence of permutations of S − x,20 and adjoin x to the front of each one. is yields, for each x in S, the sequence of permutations of S that begin with x. Combining these sequences for all x gives all the permutations of S:21 (define (permutations s) (if (null? s) ; empty set? (list nil) ; sequence containing empty set (flatmap (lambda (x) (map (lambda (p) (cons x p)) (permutations (remove x s)))) s))) 20 e set S − x is the set of all elements of S, excluding x. 21 Semicolons in Scheme code are used to introduce comments. Everything from the semicolon to the end of the line is ignored by the interpreter. In this book we don’t use many comments; we try to make our programs self-documenting by using descriptive names. 168 Notice how this strategy reduces the problem of generating permuta- tions of S to the problem of generating the permutations of sets with fewer elements than S. In the terminal case, we work our way down to the empty list, which represents a set of no elements. For this, we gen- erate (list nil), which is a sequence with one item, namely the set with no elements. e remove procedure used in permutations returns all the items in a given sequence except for a given item. is can be expressed as a simple ﬁlter: (define (remove item sequence) (filter (lambda (x) (not (= x item))) sequence)) Exercise 2.40: Deﬁne a procedure unique-pairs that, given an integer n, generates the sequence of pairs (i, j) with 1 ≤ j < i ≤ n. Use unique-pairs to simplify the deﬁnition of prime-sum-pairs given above. Exercise 2.41: Write a procedure to ﬁnd all ordered triples of distinct positive integers i, j, and k less than or equal to a given integer n that sum to a given integer s. Exercise 2.42: e “eight-queens puzzle” asks how to place eight queens on a chessboard so that no queen is in check from any other (i.e., no two queens are in the same row, col- umn, or diagonal). One possible solution is shown in Figure 2.8. One way to solve the puzzle is to work across the board, placing a queen in each column. Once we have placed k − 1 queens, we must place the k th queen in a position where it does not check any of the queens already on the board. We can formulate this approach recursively: Assume that we 169 Figure 2.8: A solution to the eight-queens puzzle. have already generated the sequence of all possible ways to place k − 1 queens in the ﬁrst k − 1 columns of the board. For each of these ways, generate an extended set of posi- tions by placing a queen in each row of the k th column. Now ﬁlter these, keeping only the positions for which the queen in the k th column is safe with respect to the other queens. is produces the sequence of all ways to place k queens in the ﬁrst k columns. By continuing this process, we will produce not only one solution, but all solutions to the puzzle. We implement this solution as a procedure queens, which returns a sequence of all solutions to the problem of plac- ing n queens on an n × n chessboard. queens has an inter- nal procedure queen-cols that returns the sequence of all ways to place queens in the ﬁrst k columns of the board. 170 (define (queens board-size) (define (queen-cols k) (if (= k 0) (list empty-board) (filter (lambda (positions) (safe? k positions)) (flatmap (lambda (rest-of-queens) (map (lambda (new-row) (adjoin-position new-row k rest-of-queens)) (enumerate-interval 1 board-size))) (queen-cols (- k 1)))))) (queen-cols board-size)) In this procedure rest-of-queens is a way to place k − 1 queens in the ﬁrst k −1 columns, and new-row is a proposed row in which to place the queen for the k th column. Com- plete the program by implementing the representation for sets of board positions, including the procedure adjoin- position, which adjoins a new row-column position to a set of positions, and empty-board, which represents an empty set of positions. You must also write the procedure safe?, which determines for a set of positions, whether the queen in the k th column is safe with respect to the others. (Note that we need only check whether the new queen is safe— the other queens are already guaranteed safe with respect to each other.) Exercise 2.43: Louis Reasoner is having a terrible time do- ing Exercise 2.42. His queens procedure seems to work, but it runs extremely slowly. (Louis never does manage to wait 171 long enough for it to solve even the 6 × 6 case.) When Louis asks Eva Lu Ator for help, she points out that he has inter- changed the order of the nested mappings in the flatmap, writing it as (flatmap (lambda (new-row) (map (lambda (rest-of-queens) (adjoin-position new-row k rest-of-queens)) (queen-cols (- k 1)))) (enumerate-interval 1 board-size)) Explain why this interchange makes the program run slowly. Estimate how long it will take Louis’s program to solve the eight-queens puzzle, assuming that the program in Exercise 2.42 solves the puzzle in time T . 2.2.4 Example: A Picture Language is section presents a simple language for drawing pictures that il- lustrates the power of data abstraction and closure, and also exploits higher-order procedures in an essential way. e language is designed to make it easy to experiment with paerns such as the ones in Fig- ure 2.9, which are composed of repeated elements that are shied and scaled.22 In this language, the data objects being combined are repre- sented as procedures rather than as list structure. Just as cons, which satisﬁes the closure property, allowed us to easily build arbitrarily com- plicated list structure, the operations in this language, which also sat- 22 e picture language is based on the language Peter Henderson created to construct images like M.C. Escher’s “Square Limit” woodcut (see Henderson 1982). e woodcut incorporates a repeated scaled paern, similar to the arrangements drawn using the square-limit procedure in this section. 172 Figure 2.9: Designs generated with the picture language. isfy the closure property, allow us to easily build arbitrarily complicated paerns. The picture language When we began our study of programming in Section 1.1, we empha- sized the importance of describing a language by focusing on the lan- guage’s primitives, its means of combination, and its means of abstrac- tion. We’ll follow that framework here. Part of the elegance of this picture language is that there is only one kind of element, called a painter. A painter draws an image that is shied and scaled to ﬁt within a designated parallelogram-shaped frame. For example, there’s a primitive painter we’ll call wave that makes a crude line drawing, as shown in Figure 2.10. e actual shape of the drawing depends on the frame—all four images in ﬁgure 2.10 are produced by the 173 Figure 2.10: Images produced by the wave painter, with respect to four diﬀerent frames. e frames, shown with doed lines, are not part of the images. same wave painter, but with respect to four diﬀerent frames. Painters can be more elaborate than this: e primitive painter called rogers paints a picture of ’s founder, William Barton Rogers, as shown in Figure 2.11.23 e four images in ﬁgure 2.11 are drawn with respect to 23 William Barton Rogers (1804-1882) was the founder and ﬁrst president of . A geologist and talented teacher, he taught at William and Mary College and at the University of Virginia. In 1859 he moved to Boston, where he had more time for re- search, worked on a plan for establishing a “polytechnic institute,” and served as Mas- sachuses’s ﬁrst State Inspector of Gas Meters. When was established in 1861, Rogers was elected its ﬁrst president. Rogers espoused an ideal of “useful learning” that was diﬀerent from the university education of the time, with its overemphasis on the classics, which, as he wrote, “stand in the way of the broader, higher and more practical instruction and discipline of the natural and social sciences.” is education was likewise to be diﬀerent from narrow trade-school 174 the same four frames as the wave images in ﬁgure 2.10. To combine images, we use various operations that construct new painters from given painters. For example, the beside operation takes two painters and produces a new, compound painter that draws the ﬁrst education. In Rogers’s words: e world-enforced distinction between the practical and the scientiﬁc worker is uerly futile, and the whole experience of modern times has demonstrated its uer worthlessness. Rogers served as president of until 1870, when he resigned due to ill health. In 1878 the second president of , John Runkle, resigned under the pressure of a ﬁnancial crisis brought on by the Panic of 1873 and strain of ﬁghting oﬀ aempts by Harvard to take over . Rogers returned to hold the oﬃce of president until 1881. Rogers collapsed and died while addressing ’s graduating class at the commence- ment exercises of 1882. Runkle quoted Rogers’s last words in a memorial address de- livered that same year: “As I stand here today and see what the Institute is, . . . I call to mind the beginnings of science. I remember one hundred and ﬁy years ago Stephen Hales published a pamphlet on the subject of illuminating gas, in which he stated that his researches had demonstrated that 128 grains of bituminous coal – ” “Bituminous coal,” these were his last words on earth. Here he bent forward, as if consulting some notes on the table before him, then slowly regaining an erect position, threw up his hands, and was translated from the scene of his earthly labors and triumphs to “the tomorrow of death,” where the mysteries of life are solved, and the disembodied spirit ﬁnds unending satisfaction in contemplating the new and still unfathomable mysteries of the inﬁnite future. In the words of Francis A. Walker (’s third president): All his life he had borne himself most faithfully and heroically, and he died as so good a knight would surely have wished, in harness, at his post, and in the very part and act of public duty. 175 Figure 2.11: Images of William Barton Rogers, founder and ﬁrst president of , painted with respect to the same four frames as in Figure 2.10 (original image from Wikimedia Commons). painter’s image in the le half of the frame and the second painter’s im- age in the right half of the frame. Similarly, below takes two painters and produces a compound painter that draws the ﬁrst painter’s image below the second painter’s image. Some operations transform a single painter to produce a new painter. For example, flip-vert takes a painter and produces a painter that draws its image upside-down, and flip-horiz produces a painter that draws the original painter’s image le-to-right reversed. Figure 2.12 shows the drawing of a painter called wave4 that is built up in two stages starting from wave: (define wave2 (beside wave (flip-vert wave))) (define wave4 (below wave2 wave2)) 176 Figure 2.12: Creating a complex ﬁgure, starting from the wave painter of Figure 2.10. In building up a complex image in this manner we are exploiting the fact that painters are closed under the language’s means of combination. e beside or below of two painters is itself a painter; therefore, we can use it as an element in making more complex painters. As with building up list structure using cons, the closure of our data under the means of combination is crucial to the ability to create complex structures while using only a few operations. Once we can combine painters, we would like to be able to abstract typical paerns of combining painters. We will implement the painter operations as Scheme procedures. is means that we don’t need a spe- cial abstraction mechanism in the picture language: Since the means of combination are ordinary Scheme procedures, we automatically have the capability to do anything with painter operations that we can do with procedures. For example, we can abstract the paern in wave4 as (define (flipped-pairs painter) (let ((painter2 (beside painter (flip-vert painter)))) (below painter2 painter2))) and deﬁne wave4 as an instance of this paern: (define wave4 (flipped-pairs wave)) 177 up- up- right-split split split corner-split n--1 n--1 n--1 n--1 identity right-split right-split n--1 identity n--1 right-split n--1 right-split n corner-split n Figure 2.13: Recursive plans for right-split and corner-split. We can also deﬁne recursive operations. Here’s one that makes painters split and branch towards the right as shown in Figure 2.13 and Figure 2.14: (define (right-split painter n) (if (= n 0) painter (let ((smaller (right-split painter (- n 1)))) (beside painter (below smaller smaller))))) We can produce balanced paerns by branching upwards as well as towards the right (see exercise Exercise 2.44 and ﬁgures Figure 2.13 and Figure 2.14): (define (corner-split painter n) (if (= n 0) painter 178 (let ((up (up-split painter (- n 1))) (right (right-split painter (- n 1)))) (let ((top-left (beside up up)) (bottom-right (below right right)) (corner (corner-split painter (- n 1)))) (beside (below painter top-left) (below bottom-right corner)))))) By placing four copies of a corner-split appropriately, we obtain a paern called square-limit, whose application to wave and rogers is shown in Figure 2.9: (define (square-limit painter n) (let ((quarter (corner-split painter n))) (let ((half (beside (flip-horiz quarter) quarter))) (below (flip-vert half) half)))) Exercise 2.44: Deﬁne the procedure up-split used by corner- split. It is similar to right-split, except that it switches the roles of below and beside. Higher-order operations In addition to abstracting paerns of combining painters, we can work at a higher level, abstracting paerns of combining painter operations. at is, we can view the painter operations as elements to manipulate and can write means of combination for these elements—procedures that take painter operations as arguments and create new painter oper- ations. For example, flipped-pairs and square-limit each arrange four copies of a painter’s image in a square paern; they diﬀer only in how 179 (right-split wave 4) (right-split rogers 4) (corner-split wave 4) (corner-split rogers 4) Figure 2.14: e recursive operations right-split and corner-split applied to the painters wave and rogers. Combining four corner-split ﬁgures produces symmet- ric square-limit designs as shown in Figure 2.9. 180 they orient the copies. One way to abstract this paern of painter com- bination is with the following procedure, which takes four one-argument painter operations and produces a painter operation that transforms a given painter with those four operations and arranges the results in a square. tl, tr, bl, and br are the transformations to apply to the top le copy, the top right copy, the boom le copy, and the boom right copy, respectively. (define (square-of-four tl tr bl br) (lambda (painter) (let ((top (beside (tl painter) (tr painter))) (bottom (beside (bl painter) (br painter)))) (below bottom top)))) en flipped-pairs can be deﬁned in terms of square-of-four as follows:24 (define (flipped-pairs painter) (let ((combine4 (square-of-four identity flip-vert identity flip-vert))) (combine4 painter))) and square-limit can be expressed as25 (define (square-limit painter n) (let ((combine4 (square-of-four flip-horiz identity rotate180 flip-vert))) 24 Equivalently, we could write (define flipped-pairs (square-of-four identity flip-vert identity flip-vert)) 25 rotate180 rotates a painter by 180 degrees (see Exercise 2.50). Instead of ro- tate180 we could say (compose flip-vert flip-horiz), using the compose pro- cedure from Exercise 1.42. 181 (combine4 (corner-split painter n)))) Exercise 2.45: right-split and up-split can be expressed as instances of a general spliing operation. Deﬁne a pro- cedure split with the property that evaluating (define right-split (split beside below)) (define up-split (split below beside)) produces procedures right-split and up-split with the same behaviors as the ones already deﬁned. Frames Before we can show how to implement painters and their means of com- bination, we must ﬁrst consider frames. A frame can be described by three vectors—an origin vector and two edge vectors. e origin vector speciﬁes the oﬀset of the frame’s origin from some absolute origin in the plane, and the edge vectors specify the oﬀsets of the frame’s cor- ners from its origin. If the edges are perpendicular, the frame will be rectangular. Otherwise the frame will be a more general parallelogram. Figure 2.15 shows a frame and its associated vectors. In accordance with data abstraction, we need not be speciﬁc yet about how frames are represented, other than to say that there is a constructor make-frame, which takes three vectors and produces a frame, and three correspond- ing selectors origin-frame, edge1-frame, and edge2-frame (see Exer- cise 2.47). We will use coordinates in the unit square (0 ≤ x , y ≤ 1) to specify images. With each frame, we associate a frame coordinate map, which will be used to shi and scale images to ﬁt the frame. e map trans- forms the unit square into the frame by mapping the vector v = (x , y) 182 frame frame edge2 edge1 vector vector frame origin (0, 0) point on vector display screen Figure 2.15: A frame is described by three vectors — an origin and two edges. to the vector sum Origin(Frame) + x · Edge1 (Frame) + y · Edge2 (Frame). For example, (0, 0) is mapped to the origin of the frame, (1, 1) to the vertex diagonally opposite the origin, and (0.5, 0.5) to the center of the frame. We can create a frame’s coordinate map with the following pro- cedure:26 (define (frame-coord-map frame) (lambda (v) (add-vect (origin-frame frame) 26 frame-coord-map uses the vector operations described in Exercise 2.46 below, which we assume have been implemented using some representation for vectors. Be- cause of data abstraction, it doesn’t maer what this vector representation is, so long as the vector operations behave correctly. 183 (add-vect (scale-vect (xcor-vect v) (edge1-frame frame)) (scale-vect (ycor-vect v) (edge2-frame frame)))))) Observe that applying frame-coord-map to a frame returns a procedure that, given a vector, returns a vector. If the argument vector is in the unit square, the result vector will be in the frame. For example, ((frame-coord-map a-frame) (make-vect 0 0)) returns the same vector as (origin-frame a-frame) Exercise 2.46: A two-dimensional vector v running from the origin to a point can be represented as a pair consisting of an x-coordinate and a y-coordinate. Implement a data abstraction for vectors by giving a constructor make-vect and corresponding selectors xcor-vect and ycor-vect. In terms of your selectors and constructor, implement proce- dures add-vect, sub-vect, and scale-vect that perform the operations vector addition, vector subtraction, and mul- tiplying a vector by a scalar: (x 1 , y1 ) + (x 2 , y2 ) = (x 1 + x 2 , y1 + y2 ), (x 1 , y1 ) − (x 2 , y2 ) = (x 1 − x 2 , y1 − y2 ), s · (x , y) = (sx , sy). Exercise 2.47: Here are two possible constructors for frames: (define (make-frame origin edge1 edge2) (list origin edge1 edge2)) (define (make-frame origin edge1 edge2) (cons origin (cons edge1 edge2))) For each constructor supply the appropriate selectors to produce an implementation for frames. 184 Painters A painter is represented as a procedure that, given a frame as argument, draws a particular image shied and scaled to ﬁt the frame. at is to say, if p is a painter and f is a frame, then we produce p’s image in f by calling p with f as argument. e details of how primitive painters are implemented depend on the particular characteristics of the graphics system and the type of im- age to be drawn. For instance, suppose we have a procedure draw-line that draws a line on the screen between two speciﬁed points. en we can create painters for line drawings, such as the wave painter in Figure 2.10, from lists of line segments as follows:27 (define (segments->painter segment-list) (lambda (frame) (for-each (lambda (segment) (draw-line ((frame-coord-map frame) (start-segment segment)) ((frame-coord-map frame) (end-segment segment)))) segment-list))) e segments are given using coordinates with respect to the unit square. For each segment in the list, the painter transforms the segment end- points with the frame coordinate map and draws a line between the transformed points. Representing painters as procedures erects a powerful abstraction barrier in the picture language. We can create and intermix all sorts of 27 segments->painter uses the representation for line segments described in Exer- cise 2.48 below. It also uses the for-each procedure described in Exercise 2.23. 185 primitive painters, based on a variety of graphics capabilities. e de- tails of their implementation do not maer. Any procedure can serve as a painter, provided that it takes a frame as argument and draws some- thing scaled to ﬁt the frame.28 Exercise 2.48: A directed line segment in the plane can be represented as a pair of vectors—the vector running from the origin to the start-point of the segment, and the vector running from the origin to the end-point of the segment. Use your vector representation from Exercise 2.46 to de- ﬁne a representation for segments with a constructor make- segment and selectors start-segment and end-segment. Exercise 2.49: Use segments->painter to deﬁne the fol- lowing primitive painters: a. e painter that draws the outline of the designated frame. b. e painter that draws an “X” by connecting opposite corners of the frame. c. e painter that draws a diamond shape by connect- ing the midpoints of the sides of the frame. d. e wave painter. 28 For example, the rogers painter of Figure 2.11 was constructed from a gray-level image. For each point in a given frame, the rogers painter determines the point in the image that is mapped to it under the frame coordinate map, and shades it accordingly. By allowing diﬀerent types of painters, we are capitalizing on the abstract data idea discussed in Section 2.1.3, where we argued that a rational-number representation could be anything at all that satisﬁes an appropriate condition. Here we’re using the fact that a painter can be implemented in any way at all, so long as it draws something in the designated frame. Section 2.1.3 also showed how pairs could be implemented as procedures. Painters are our second example of a procedural representation for data. 186 Transforming and combining painters An operation on painters (such as flip-vert or beside) works by cre- ating a painter that invokes the original painters with respect to frames derived from the argument frame. us, for example, flip-vert doesn’t have to know how a painter works in order to ﬂip it—it just has to know how to turn a frame upside down: e ﬂipped painter just uses the orig- inal painter, but in the inverted frame. Painter operations are based on the procedure transform-painter, which takes as arguments a painter and information on how to trans- form a frame and produces a new painter. e transformed painter, when called on a frame, transforms the frame and calls the original painter on the transformed frame. e arguments to transform-painter are points (represented as vectors) that specify the corners of the new frame: When mapped into the frame, the ﬁrst point speciﬁes the new frame’s origin and the other two specify the ends of its edge vectors. us, arguments within the unit square specify a frame contained within the original frame. (define (transform-painter painter origin corner1 corner2) (lambda (frame) (let ((m (frame-coord-map frame))) (let ((new-origin (m origin))) (painter (make-frame new-origin (sub-vect (m corner1) new-origin) (sub-vect (m corner2) new-origin))))))) Here’s how to ﬂip painter images vertically: (define (flip-vert painter) (transform-painter painter (make-vect 0.0 1.0) ; new origin 187 (make-vect 1.0 1.0) ; new end of edge1 (make-vect 0.0 0.0))) ; new end of edge2 Using transform-painter, we can easily deﬁne new transformations. For example, we can deﬁne a painter that shrinks its image to the upper- right quarter of the frame it is given: (define (shrink-to-upper-right painter) (transform-painter painter (make-vect 0.5 0.5) (make-vect 1.0 0.5) (make-vect 0.5 1.0))) Other transformations rotate images counterclockwise by 90 degrees29 (define (rotate90 painter) (transform-painter painter (make-vect 1.0 0.0) (make-vect 1.0 1.0) (make-vect 0.0 0.0))) or squash images towards the center of the frame:30 (define (squash-inwards painter) (transform-painter painter (make-vect 0.0 0.0) (make-vect 0.65 0.35) (make-vect 0.35 0.65))) Frame transformation is also the key to deﬁning means of combining two or more painters. e beside procedure, for example, takes two painters, transforms them to paint in the le and right halves of an argument frame respectively, and produces a new, compound painter. 29 rotate90 is a pure rotation only for square frames, because it also stretches and shrinks the image to ﬁt into the rotated frame. 30 e diamond-shaped images in Figure 2.10 and Figure 2.11 were created with squash-inwards applied to wave and rogers. 188 When the compound painter is given a frame, it calls the ﬁrst trans- formed painter to paint in the le half of the frame and calls the second transformed painter to paint in the right half of the frame: (define (beside painter1 painter2) (let ((split-point (make-vect 0.5 0.0))) (let ((paint-left (transform-painter painter1 (make-vect 0.0 0.0) split-point (make-vect 0.0 1.0))) (paint-right (transform-painter painter2 split-point (make-vect 1.0 0.0) (make-vect 0.5 1.0)))) (lambda (frame) (paint-left frame) (paint-right frame))))) Observe how the painter data abstraction, and in particular the repre- sentation of painters as procedures, makes beside easy to implement. e beside procedure need not know anything about the details of the component painters other than that each painter will draw something in its designated frame. Exercise 2.50: Deﬁne the transformation flip-horiz, which ﬂips painters horizontally, and transformations that rotate painters counterclockwise by 180 degrees and 270 degrees. Exercise 2.51: Deﬁne the below operation for painters. below takes two painters as arguments. e resulting painter, given 189 a frame, draws with the ﬁrst painter in the boom of the frame and with the second painter in the top. Deﬁne below in two diﬀerent ways—ﬁrst by writing a procedure that is analogous to the beside procedure given above, and again in terms of beside and suitable rotation operations (from Exercise 2.50). Levels of language for robust design e picture language exercises some of the critical ideas we’ve intro- duced about abstraction with procedures and data. e fundamental data abstractions, painters, are implemented using procedural represen- tations, which enables the language to handle diﬀerent basic drawing capabilities in a uniform way. e means of combination satisfy the closure property, which permits us to easily build up complex designs. Finally, all the tools for abstracting procedures are available to us for abstracting means of combination for painters. We have also obtained a glimpse of another crucial idea about lan- guages and program design. is is the approach of stratiﬁed design, the notion that a complex system should be structured as a sequence of levels that are described using a sequence of languages. Each level is constructed by combining parts that are regarded as primitive at that level, and the parts constructed at each level are used as primitives at the next level. e language used at each level of a stratiﬁed design has primitives, means of combination, and means of abstraction appropriate to that level of detail. Stratiﬁed design pervades the engineering of complex systems. For example, in computer engineering, resistors and transistors are com- bined (and described using a language of analog circuits) to produce parts such as and-gates and or-gates, which form the primitives of a 190 language for digital-circuit design.31 ese parts are combined to build processors, bus structures, and memory systems, which are in turn com- bined to form computers, using languages appropriate to computer ar- chitecture. Computers are combined to form distributed systems, using languages appropriate for describing network interconnections, and so on. As a tiny example of stratiﬁcation, our picture language uses prim- itive elements (primitive painters) that are created using a language that speciﬁes points and lines to provide the lists of line segments for segments->painter, or the shading details for a painter like rogers. e bulk of our description of the picture language focused on com- bining these primitives, using geometric combiners such as beside and below. We also worked at a higher level, regarding beside and below as primitives to be manipulated in a language whose operations, such as square-of-four, capture common paerns of combining geometric combiners. Stratiﬁed design helps make programs robust, that is, it makes it likely that small changes in a speciﬁcation will require correspondingly small changes in the program. For instance, suppose we wanted to change the image based on wave shown in Figure 2.9. We could work at the lowest level to change the detailed appearance of the wave element; we could work at the middle level to change the way corner-split replicates the wave; we could work at the highest level to change how square-limit arranges the four copies of the corner. In general, each level of a stratiﬁed design provides a diﬀerent vocabulary for express- ing the characteristics of the system, and a diﬀerent kind of ability to change it. 31 Section 3.3.4 describes one such language. 191 Exercise 2.52: Make changes to the square limit of wave shown in Figure 2.9 by working at each of the levels de- scribed above. In particular: a. Add some segments to the primitive wave painter of Exercise 2.49 (to add a smile, for example). b. Change the paern constructed by corner-split (for example, by using only one copy of the up-split and right-split images instead of two). c. Modify the version of square-limit that uses square- of-four so as to assemble the corners in a diﬀerent paern. (For example, you might make the big Mr. Rogers look outward from each corner of the square.) 2.3 Symbolic Data All the compound data objects we have used so far were constructed ul- timately from numbers. In this section we extend the representational capability of our language by introducing the ability to work with arbi- trary symbols as data. 2.3.1 otation If we can form compound data using symbols, we can have lists such as (a b c d) (23 45 17) ((Norah 12) (Molly 9) (Anna 7) (Lauren 6) (Charlotte 4)) Lists containing symbols can look just like the expressions of our lan- guage: 192 (* (+ 23 45) (+ x 9)) (define (fact n) (if (= n 1) 1 (* n (fact (- n 1))))) In order to manipulate symbols we need a new element in our language: the ability to quote a data object. Suppose we want to construct the list (a b). We can’t accomplish this with (list a b), because this expres- sion constructs a list of the values of a and b rather than the symbols themselves. is issue is well known in the context of natural languages, where words and sentences may be regarded either as semantic entities or as character strings (syntactic entities). e common practice in nat- ural languages is to use quotation marks to indicate that a word or a sentence is to be treated literally as a string of characters. For instance, the ﬁrst leer of “John” is clearly “J.” If we tell somebody “say your name aloud,” we expect to hear that person’s name. However, if we tell somebody “say ‘your name’ aloud,” we expect to hear the words “your name.” Note that we are forced to nest quotation marks to describe what somebody else might say.32 We can follow this same practice to identify lists and symbols that are to be treated as data objects rather than as expressions to be evalu- 32 Allowing quotation in a language wreaks havoc with the ability to reason about the language in simple terms, because it destroys the notion that equals can be sub- stituted for equals. For example, three is one plus two, but the word “three” is not the phrase “one plus two.” otation is powerful because it gives us a way to build expres- sions that manipulate other expressions (as we will see when we write an interpreter in Chapter 4). But allowing statements in a language that talk about other statements in that language makes it very diﬃcult to maintain any coherent principle of what “equals can be substituted for equals” should mean. For example, if we know that the evening star is the morning star, then from the statement “the evening star is Venus” we can deduce “the morning star is Venus.” However, given that “John knows that the evening star is Venus” we cannot infer that “John knows that the morning star is Venus.” 193 ated. However, our format for quoting diﬀers from that of natural lan- guages in that we place a quotation mark (traditionally, the single quote symbol ') only at the beginning of the object to be quoted. We can get away with this in Scheme syntax because we rely on blanks and paren- theses to delimit objects. us, the meaning of the single quote character is to quote the next object.33 Now we can distinguish between symbols and their values: (define a 1) (define b 2) (list a b) (1 2) (list 'a 'b) (a b) (list 'a b) (a 2) otation also allows us to type in compound objects, using the con- ventional printed representation for lists:34 33 e single quote is diﬀerent from the double quote we have been using to enclose character strings to be printed. Whereas the single quote can be used to denote lists or symbols, the double quote is used only with character strings. In this book, the only use for character strings is as items to be printed. 34 Strictly, our use of the quotation mark violates the general rule that all compound expressions in our language should be delimited by parentheses and look like lists. We can recover this consistency by introducing a special form quote, which serves the same purpose as the quotation mark. us, we would type (quote a) instead of 'a, and we would type (quote (a b c)) instead of '(a b c). is is precisely how the interpreter works. e quotation mark is just a single-character abbreviation for wrapping the next complete expression with quote to form (quote ⟨ expression ⟩). is is important because it maintains the principle that any expression seen by the interpreter can be manipulated as a data object. For instance, we could construct the expression (car '(a b c)), which is the same as (car (quote (a b c))), by evaluating (list 'car (list 'quote '(a b c))). 194 (car '(a b c)) a (cdr '(a b c)) (b c) In keeping with this, we can obtain the empty list by evaluating '(), and thus dispense with the variable nil. One additional primitive used in manipulating symbols is eq?, which takes two symbols as arguments and tests whether they are the same.35 Using eq?, we can implement a useful procedure called memq. is takes two arguments, a symbol and a list. If the symbol is not contained in the list (i.e., is not eq? to any item in the list), then memq returns false. Other- wise, it returns the sublist of the list beginning with the ﬁrst occurrence of the symbol: (define (memq item x) (cond ((null? x) false) ((eq? item (car x)) x) (else (memq item (cdr x))))) For example, the value of (memq 'apple '(pear banana prune)) is false, whereas the value of (memq 'apple '(x (apple sauce) y apple pear)) is (apple pear). Exercise 2.53: What would the interpreter print in response to evaluating each of the following expressions? 35 We can consider two symbols to be “the same” if they consist of the same characters in the same order. Such a deﬁnition skirts a deep issue that we are not yet ready to address: the meaning of “sameness” in a programming language. We will return to this in Chapter 3 (Section 3.1.3). 195 (list 'a 'b 'c) (list (list 'george)) (cdr '((x1 x2) (y1 y2))) (cadr '((x1 x2) (y1 y2))) (pair? (car '(a short list))) (memq 'red '((red shoes) (blue socks))) (memq 'red '(red shoes blue socks)) Exercise 2.54: Two lists are said to be equal? if they con- tain equal elements arranged in the same order. For exam- ple, (equal? '(this is a list) '(this is a list)) is true, but (equal? '(this is a list) '(this (is a) list)) is false. To be more precise, we can deﬁne equal? recur- sively in terms of the basic eq? equality of symbols by say- ing that a and b are equal? if they are both symbols and the symbols are eq?, or if they are both lists such that (car a) is equal? to (car b) and (cdr a) is equal? to (cdr b). Using this idea, implement equal? as a procedure.36 Exercise 2.55: Eva Lu Ator types to the interpreter the ex- pression 36 In practice, programmers use equal? to compare lists that contain numbers as well as symbols. Numbers are not considered to be symbols. e question of whether two numerically equal numbers (as tested by =) are also eq? is highly implementation- dependent. A beer deﬁnition of equal? (such as the one that comes as a primitive in Scheme) would also stipulate that if a and b are both numbers, then a and b are equal? if they are numerically equal. 196 (car ''abracadabra) To her surprise, the interpreter prints back quote. Explain. 2.3.2 Example: Symbolic Diﬀerentiation As an illustration of symbol manipulation and a further illustration of data abstraction, consider the design of a procedure that performs sym- bolic diﬀerentiation of algebraic expressions. We would like the proce- dure to take as arguments an algebraic expression and a variable and to return the derivative of the expression with respect to the variable. For example, if the arguments to the procedure are ax 2 + bx + c and x, the procedure should return 2ax + b. Symbolic diﬀerentiation is of special historical signiﬁcance in Lisp. It was one of the motivating examples behind the development of a computer language for symbol manipula- tion. Furthermore, it marked the beginning of the line of research that led to the development of powerful systems for symbolic mathematical work, which are currently being used by a growing number of applied mathematicians and physicists. In developing the symbolic-diﬀerentiation program, we will follow the same strategy of data abstraction that we followed in developing the rational-number system of Section 2.1.1. at is, we will ﬁrst de- ﬁne a diﬀerentiation algorithm that operates on abstract objects such as “sums,” “products,” and “variables” without worrying about how these are to be represented. Only aerward will we address the representation problem. The diﬀerentiation program with abstract data In order to keep things simple, we will consider a very simple symbolic- diﬀerentiation program that handles expressions that are built up using 197 only the operations of addition and multiplication with two arguments. Diﬀerentiation of any such expression can be carried out by applying the following reduction rules: dc = 0, for c a constant or a variable different from x , dx dx = 1, dx d (u + v ) du dv = + , dx dx dx d (uv ) dv du = u +v . dx dx dx Observe that the laer two rules are recursive in nature. at is, to ob- tain the derivative of a sum we ﬁrst ﬁnd the derivatives of the terms and add them. Each of the terms may in turn be an expression that needs to be decomposed. Decomposing into smaller and smaller pieces will eventually produce pieces that are either constants or variables, whose derivatives will be either 0 or 1. To embody these rules in a procedure we indulge in a lile wishful thinking, as we did in designing the rational-number implementation. If we had a means for representing algebraic expressions, we should be able to tell whether an expression is a sum, a product, a constant, or a variable. We should be able to extract the parts of an expression. For a sum, for example we want to be able to extract the addend (ﬁrst term) and the augend (second term). We should also be able to construct expressions from parts. Let us assume that we already have procedures to implement the following selectors, constructors, and predicates: (variable? e) Is e a variable? (same-variable? v1 v2) Are v1 and v2 the same variable? 198 (sum? e) Is e a sum? (addend e) Addend of the sum e. (augend e) Augend of the sum e. (make-sum a1 a2) Construct the sum of a1 and a2. (product? e) Is e a product? (multiplier e) Multiplier of the product e. (multiplicand e) Multiplicand of the product e. (make-product m1 m2) Construct the product of m1 and m2. Using these, and the primitive predicate number?, which identiﬁes num- bers, we can express the diﬀerentiation rules as the following procedure: (define (deriv exp var) (cond ((number? exp) 0) ((variable? exp) (if (same-variable? exp var) 1 0)) ((sum? exp) (make-sum (deriv (addend exp) var) (deriv (augend exp) var))) ((product? exp) (make-sum (make-product (multiplier exp) (deriv (multiplicand exp) var)) (make-product (deriv (multiplier exp) var) (multiplicand exp)))) (else (error "unknown expression type: DERIV" exp)))) is deriv procedure incorporates the complete diﬀerentiation algo- rithm. Since it is expressed in terms of abstract data, it will work no maer how we choose to represent algebraic expressions, as long as we design a proper set of selectors and constructors. is is the issue we must address next. 199 Representing algebraic expressions We can imagine many ways to use list structure to represent algebraic expressions. For example, we could use lists of symbols that mirror the usual algebraic notation, representing ax + b as the list (a * x + b). However, one especially straightforward choice is to use the same parenthesized preﬁx notation that Lisp uses for combinations; that is, to represent ax + b as (+ (* a x) b). en our data representation for the diﬀerentiation problem is as follows: • e variables are symbols. ey are identiﬁed by the primitive predicate symbol?: (define (variable? x) (symbol? x)) • Two variables are the same if the symbols representing them are eq?: (define (same-variable? v1 v2) (and (variable? v1) (variable? v2) (eq? v1 v2))) • Sums and products are constructed as lists: (define (make-sum a1 a2) (list '+ a1 a2)) (define (make-product m1 m2) (list '* m1 m2)) • A sum is a list whose ﬁrst element is the symbol +: (define (sum? x) (and (pair? x) (eq? (car x) '+))) • e addend is the second item of the sum list: (define (addend s) (cadr s)) 200 • e augend is the third item of the sum list: (define (augend s) (caddr s)) • A product is a list whose ﬁrst element is the symbol *: (define (product? x) (and (pair? x) (eq? (car x) '*))) • e multiplier is the second item of the product list: (define (multiplier p) (cadr p)) • e multiplicand is the third item of the product list: (define (multiplicand p) (caddr p)) us, we need only combine these with the algorithm as embodied by deriv in order to have a working symbolic-diﬀerentiation program. Let us look at some examples of its behavior: (deriv '(+ x 3) 'x) (+ 1 0) (deriv '(* x y) 'x) (+ (* x 0) (* 1 y)) (deriv '(* (* x y) (+ x 3)) 'x) (+ (* (* x y) (+ 1 0)) (* (+ (* x 0) (* 1 y)) (+ x 3))) e program produces answers that are correct; however, they are un- simpliﬁed. It is true that d (xy ) = x · 0 + 1 · y, dx 201 but we would like the program to know that x · 0 = 0, 1 · y = y, and 0 + y = y. e answer for the second example should have been simply y. As the third example shows, this becomes a serious issue when the expressions are complex. Our diﬃculty is much like the one we encountered with the rational- number implementation: we haven’t reduced answers to simplest form. To accomplish the rational-number reduction, we needed to change only the constructors and the selectors of the implementation. We can adopt a similar strategy here. We won’t change deriv at all. Instead, we will change make-sum so that if both summands are numbers, make-sum will add them and return their sum. Also, if one of the summands is 0, then make-sum will return the other summand. (define (make-sum a1 a2) (cond ((=number? a1 0) a2) ((=number? a2 0) a1) ((and (number? a1) (number? a2)) (+ a1 a2)) (else (list '+ a1 a2)))) is uses the procedure =number?, which checks whether an expression is equal to a given number: (define (=number? exp num) (and (number? exp) (= exp num))) Similarly, we will change make-product to build in the rules that 0 times anything is 0 and 1 times anything is the thing itself: (define (make-product m1 m2) (cond ((or (=number? m1 0) (=number? m2 0)) 0) ((=number? m1 1) m2) ((=number? m2 1) m1) ((and (number? m1) (number? m2)) (* m1 m2)) (else (list '* m1 m2)))) 202 Here is how this version works on our three examples: (deriv '(+ x 3) 'x) 1 (deriv '(* x y) 'x) y (deriv '(* (* x y) (+ x 3)) 'x) (+ (* x y) (* y (+ x 3))) Although this is quite an improvement, the third example shows that there is still a long way to go before we get a program that puts ex- pressions into a form that we might agree is “simplest.” e problem of algebraic simpliﬁcation is complex because, among other reasons, a form that may be simplest for one purpose may not be for another. Exercise 2.56: Show how to extend the basic diﬀerentiator to handle more kinds of expressions. For instance, imple- ment the diﬀerentiation rule d (un ) du = nun −1 dx dx by adding a new clause to the deriv program and deﬁning appropriate procedures exponentiation?, base, exponent, and make-exponentiation. (You may use the symbol ** to denote exponentiation.) Build in the rules that anything raised to the power 0 is 1 and anything raised to the power 1 is the thing itself. Exercise 2.57: Extend the diﬀerentiation program to han- dle sums and products of arbitrary numbers of (two or more) terms. en the last example above could be expressed as (deriv '(* x y (+ x 3)) 'x) 203 Try to do this by changing only the representation for sums and products, without changing the deriv procedure at all. For example, the addend of a sum would be the ﬁrst term, and the augend would be the sum of the rest of the terms. Exercise 2.58: Suppose we want to modify the diﬀerentia- tion program so that it works with ordinary mathematical notation, in which + and * are inﬁx rather than preﬁx opera- tors. Since the diﬀerentiation program is deﬁned in terms of abstract data, we can modify it to work with diﬀerent repre- sentations of expressions solely by changing the predicates, selectors, and constructors that deﬁne the representation of the algebraic expressions on which the diﬀerentiator is to operate. a. Show how to do this in order to diﬀerentiate algebraic expressions presented in inﬁx form, such as (x + (3 * (x + (y + 2)))). To simplify the task, assume that + and * always take two arguments and that expres- sions are fully parenthesized. b. e problem becomes substantially harder if we allow standard algebraic notation, such as (x + 3 * (x + y + 2)), which drops unnecessary parentheses and assumes that multiplication is done before addition. Can you design appropriate predicates, selectors, and constructors for this notation such that our derivative program still works? 204 2.3.3 Example: Representing Sets In the previous examples we built representations for two kinds of com- pound data objects: rational numbers and algebraic expressions. In one of these examples we had the choice of simplifying (reducing) the ex- pressions at either construction time or selection time, but other than that the choice of a representation for these structures in terms of lists was straightforward. When we turn to the representation of sets, the choice of a representation is not so obvious. Indeed, there are a num- ber of possible representations, and they diﬀer signiﬁcantly from one another in several ways. Informally, a set is simply a collection of distinct objects. To give a more precise deﬁnition we can employ the method of data abstrac- tion. at is, we deﬁne “set” by specifying the operations that are to be used on sets. ese are union-set, intersection-set, element-of- set?, and adjoin-set. element-of-set? is a predicate that determines whether a given element is a member of a set. adjoin-set takes an ob- ject and a set as arguments and returns a set that contains the elements of the original set and also the adjoined element. union-set computes the union of two sets, which is the set containing each element that appears in either argument. intersection-set computes the intersec- tion of two sets, which is the set containing only elements that appear in both arguments. From the viewpoint of data abstraction, we are free to design any representation that implements these operations in a way consistent with the interpretations given above.37 37 If we want to be more formal, we can specify “consistent with the interpretations given above” to mean that the operations satisfy a collection of rules such as these: • For any set S and any object x, (element-of-set? x (adjoin-set x S)) is true (informally: “Adjoining an object to a set produces a set that contains the object”). • For any sets S and T and any object x, (element-of-set? x (union-set S T)) is 205 Sets as unordered lists One way to represent a set is as a list of its elements in which no el- ement appears more than once. e empty set is represented by the empty list. In this representation, element-of-set? is similar to the procedure memq of Section 2.3.1. It uses equal? instead of eq? so that the set elements need not be symbols: (define (element-of-set? x set) (cond ((null? set) false) ((equal? x (car set)) true) (else (element-of-set? x (cdr set))))) Using this, we can write adjoin-set. If the object to be adjoined is al- ready in the set, we just return the set. Otherwise, we use cons to add the object to the list that represents the set: (define (adjoin-set x set) (if (element-of-set? x set) set (cons x set))) For intersection-set we can use a recursive strategy. If we know how to form the intersection of set2 and the cdr of set1, we only need to decide whether to include the car of set1 in this. But this depends on whether (car set1) is also in set2. Here is the resulting procedure: (define (intersection-set set1 set2) (cond ((or (null? set1) (null? set2)) '()) ((element-of-set? (car set1) set2) (cons (car set1) (intersection-set (cdr set1) set2))) (else (intersection-set (cdr set1) set2)))) equal to (or (element-of-set? x S) (element-of-set? x T)) (informally: “e elements of (union S T) are the elements that are in S or in T”). • For any object x, (element-of-set? x '()) is false (informally: “No object is an element of the empty set”). 206 In designing a representation, one of the issues we should be concerned with is eﬃciency. Consider the number of steps required by our set operations. Since they all use element-of-set?, the speed of this oper- ation has a major impact on the eﬃciency of the set implementation as a whole. Now, in order to check whether an object is a member of a set, element-of-set? may have to scan the entire set. (In the worst case, the object turns out not to be in the set.) Hence, if the set has n elements, element-of-set? might take up to n steps. us, the number of steps required grows as Θ(n). e number of steps required by adjoin-set, which uses this operation, also grows as Θ(n). For intersection-set, which does an element-of-set? check for each element of set1, the number of steps required grows as the product of the sizes of the sets involved, or Θ(n 2 ) for two sets of size n. e same will be true of union- set. Exercise 2.59: Implement the union-set operation for the unordered-list representation of sets. Exercise 2.60: We speciﬁed that a set would be represented as a list with no duplicates. Now suppose we allow dupli- cates. For instance, the set {1, 2, 3} could be represented as the list (2 3 2 1 3 2 2). Design procedures element- of-set?, adjoin-set, union-set, and intersection-set that operate on this representation. How does the eﬃciency of each compare with the corresponding procedure for the non-duplicate representation? Are there applications for which you would use this representation in preference to the non- duplicate one? 207 Sets as ordered lists One way to speed up our set operations is to change the representation so that the set elements are listed in increasing order. To do this, we need some way to compare two objects so that we can say which is bigger. For example, we could compare symbols lexicographically, or we could agree on some method for assigning a unique number to an object and then compare the elements by comparing the corresponding numbers. To keep our discussion simple, we will consider only the case where the set elements are numbers, so that we can compare elements using > and <. We will represent a set of numbers by listing its elements in increasing order. Whereas our ﬁrst representation above allowed us to represent the set {1, 3, 6, 10} by listing the elements in any order, our new representation allows only the list (1 3 6 10). One advantage of ordering shows up in element-of-set?: In check- ing for the presence of an item, we no longer have to scan the entire set. If we reach a set element that is larger than the item we are looking for, then we know that the item is not in the set: (define (element-of-set? x set) (cond ((null? set) false) ((= x (car set)) true) ((< x (car set)) false) (else (element-of-set? x (cdr set))))) How many steps does this save? In the worst case, the item we are looking for may be the largest one in the set, so the number of steps is the same as for the unordered representation. On the other hand, if we search for items of many diﬀerent sizes we can expect that some- times we will be able to stop searching at a point near the beginning of the list and that other times we will still need to examine most of the list. On the average we should expect to have to examine about half of 208 the items in the set. us, the average number of steps required will be about n/2. is is still Θ(n) growth, but it does save us, on the average, a factor of 2 in number of steps over the previous implementation. We obtain a more impressive speedup with intersection-set. In the unordered representation this operation required Θ(n 2 ) steps, be- cause we performed a complete scan of set2 for each element of set1. But with the ordered representation, we can use a more clever method. Begin by comparing the initial elements, x1 and x2, of the two sets. If x1 equals x2, then that gives an element of the intersection, and the rest of the intersection is the intersection of the cdr-s of the two sets. Sup- pose, however, that x1 is less than x2. Since x2 is the smallest element in set2, we can immediately conclude that x1 cannot appear anywhere in set2 and hence is not in the intersection. Hence, the intersection is equal to the intersection of set2 with the cdr of set1. Similarly, if x2 is less than x1, then the intersection is given by the intersection of set1 with the cdr of set2. Here is the procedure: (define (intersection-set set1 set2) (if (or (null? set1) (null? set2)) '() (let ((x1 (car set1)) (x2 (car set2))) (cond ((= x1 x2) (cons x1 (intersection-set (cdr set1) (cdr set2)))) ((< x1 x2) (intersection-set (cdr set1) set2)) ((< x2 x1) (intersection-set set1 (cdr set2))))))) To estimate the number of steps required by this process, observe that at each step we reduce the intersection problem to computing inter- sections of smaller sets—removing the ﬁrst element from set1 or set2 209 or both. us, the number of steps required is at most the sum of the sizes of set1 and set2, rather than the product of the sizes as with the unordered representation. is is Θ(n) growth rather than Θ(n 2 )—a con- siderable speedup, even for sets of moderate size. Exercise 2.61: Give an implementation of adjoin-set us- ing the ordered representation. By analogy with element- of-set? show how to take advantage of the ordering to produce a procedure that requires on the average about half as many steps as with the unordered representation. Exercise 2.62: Give a Θ(n) implementation of union-set for sets represented as ordered lists. Sets as binary trees We can do beer than the ordered-list representation by arranging the set elements in the form of a tree. Each node of the tree holds one ele- ment of the set, called the “entry” at that node, and a link to each of two other (possibly empty) nodes. e “le” link points to elements smaller than the one at the node, and the “right” link to elements greater than the one at the node. Figure 2.16 shows some trees that represent the set {1, 3, 5, 7, 9, 11}. e same set may be represented by a tree in a number of diﬀerent ways. e only thing we require for a valid representation is that all elements in the le subtree be smaller than the node entry and that all elements in the right subtree be larger. e advantage of the tree representation is this: Suppose we want to check whether a number x is contained in a set. We begin by comparing x with the entry in the top node. If x is less than this, we know that we need only search the le subtree; if x is greater, we need only search the right subtree. Now, if the tree is “balanced,” each of these subtrees 210 7 3 5 3 9 1 7 3 9 1 5 11 5 9 1 7 11 11 Figure 2.16: Various binary trees that represent the set {1, 3, 5, 7, 9, 11}. will be about half the size of the original. us, in one step we have reduced the problem of searching a tree of size n to searching a tree of size n/2. Since the size of the tree is halved at each step, we should expect that the number of steps needed to search a tree of size n grows as Θ(log n).38 For large sets, this will be a signiﬁcant speedup over the previous representations. We can represent trees by using lists. Each node will be a list of three items: the entry at the node, the le subtree, and the right subtree. A le or a right subtree of the empty list will indicate that there is no subtree connected there. We can describe this representation by the following procedures:39 38 Halving the size of the problem at each step is the distinguishing characteristic of logarithmic growth, as we saw with the fast-exponentiation algorithm of Section 1.2.4 and the half-interval search method of Section 1.3.3. 39 We are representing sets in terms of trees, and trees in terms of lists—in eﬀect, a data abstraction built upon a data abstraction. We can regard the procedures entry, left-branch, right-branch, and make-tree as a way of isolating the abstraction of a “binary tree” from the particular way we might wish to represent such a tree in terms of list structure. 211 (define (entry tree) (car tree)) (define (left-branch tree) (cadr tree)) (define (right-branch tree) (caddr tree)) (define (make-tree entry left right) (list entry left right)) Now we can write the element-of-set? procedure using the strategy described above: (define (element-of-set? x set) (cond ((null? set) false) ((= x (entry set)) true) ((< x (entry set)) (element-of-set? x (left-branch set))) ((> x (entry set)) (element-of-set? x (right-branch set))))) Adjoining an item to a set is implemented similarly and also requires Θ(log n) steps. To adjoin an item x, we compare x with the node en- try to determine whether x should be added to the right or to the le branch, and having adjoined x to the appropriate branch we piece this newly constructed branch together with the original entry and the other branch. If x is equal to the entry, we just return the node. If we are asked to adjoin x to an empty tree, we generate a tree that has x as the entry and empty right and le branches. Here is the procedure: (define (adjoin-set x set) (cond ((null? set) (make-tree x '() '())) ((= x (entry set)) set) ((< x (entry set)) (make-tree (entry set) (adjoin-set x (left-branch set)) (right-branch set))) ((> x (entry set)) 212 (make-tree (entry set) (left-branch set) (adjoin-set x (right-branch set)))))) e above claim that searching the tree can be performed in a logarith- mic number of steps rests on the assumption that the tree is “balanced,” i.e., that the le and the right subtree of every tree have approximately the same number of elements, so that each subtree contains about half the elements of its parent. But how can we be certain that the trees we construct will be balanced? Even if we start with a balanced tree, adding elements with adjoin-set may produce an unbalanced result. Since the position of a newly adjoined element depends on how the element com- pares with the items already in the set, we can expect that if we add ele- ments “randomly” the tree will tend to be balanced on the average. But this is not a guarantee. For example, if we start with an empty set and adjoin the numbers 1 through 7 in sequence we end up with the highly unbalanced tree shown in Figure 2.17. In this tree all the le subtrees are empty, so it has no advantage over a simple ordered list. One way to solve this problem is to deﬁne an operation that transforms an arbitrary tree into a balanced tree with the same elements. en we can perform this transformation aer every few adjoin-set operations to keep our set in balance. ere are also other ways to solve this problem, most of which involve designing new data structures for which searching and insertion both can be done in Θ(log n) steps.40 Exercise 2.63: Each of the following two procedures con- verts a binary tree to a list. (define (tree->list-1 tree) (if (null? tree) 40 Examples of such structures include B-trees and red-black trees. ere is a large literature on data structures devoted to this problem. See Cormen et al. 1990. 213 1 2 3 4 5 6 7 Figure 2.17: Unbalanced tree produced by adjoining 1 through 7 in sequence. '() (append (tree->list-1 (left-branch tree)) (cons (entry tree) (tree->list-1 (right-branch tree)))))) (define (tree->list-2 tree) (define (copy-to-list tree result-list) (if (null? tree) result-list (copy-to-list (left-branch tree) (cons (entry tree) (copy-to-list (right-branch tree) result-list))))) (copy-to-list tree '())) a. Do the two procedures produce the same result for every tree? If not, how do the results diﬀer? What lists 214 do the two procedures produce for the trees in Figure 2.16? b. Do the two procedures have the same order of growth in the number of steps required to convert a balanced tree with n elements to a list? If not, which one grows more slowly? Exercise 2.64: e following procedure list->tree con- verts an ordered list to a balanced binary tree. e helper procedure partial-tree takes as arguments an integer n and list of at least n elements and constructs a balanced tree containing the ﬁrst n elements of the list. e result re- turned by partial-tree is a pair (formed with cons) whose car is the constructed tree and whose cdr is the list of ele- ments not included in the tree. (define (list->tree elements) (car (partial-tree elements (length elements)))) (define (partial-tree elts n) (if (= n 0) (cons '() elts) (let ((left-size (quotient (- n 1) 2))) (let ((left-result (partial-tree elts left-size))) (let ((left-tree (car left-result)) (non-left-elts (cdr left-result)) (right-size (- n (+ left-size 1)))) (let ((this-entry (car non-left-elts)) (right-result (partial-tree (cdr non-left-elts) right-size))) 215 (let ((right-tree (car right-result)) (remaining-elts (cdr right-result))) (cons (make-tree this-entry left-tree right-tree) remaining-elts)))))))) a. Write a short paragraph explaining as clearly as you can how partial-tree works. Draw the tree produced by list->tree for the list (1 3 5 7 9 11). b. What is the order of growth in the number of steps re- quired by list->tree to convert a list of n elements? Exercise 2.65: Use the results of Exercise 2.63 and Exer- cise 2.64 to give Θ(n) implementations of union-set and intersection-set for sets implemented as (balanced) bi- nary trees.41 Sets and information retrieval We have examined options for using lists to represent sets and have seen how the choice of representation for a data object can have a large impact on the performance of the programs that use the data. Another reason for concentrating on sets is that the techniques discussed here appear again and again in applications involving information retrieval. Consider a data base containing a large number of individual records, such as the personnel ﬁles for a company or the transactions in an ac- counting system. A typical data-management system spends a large 41 Exercise 2.63 through Exercise 2.65 are due to Paul Hilﬁnger. 216 amount of time accessing or modifying the data in the records and therefore requires an eﬃcient method for accessing records. is is done by identifying a part of each record to serve as an identifying key. A key can be anything that uniquely identiﬁes the record. For a personnel ﬁle, it might be an employee’s number. For an accounting system, it might be a transaction number. Whatever the key is, when we deﬁne the record as a data structure we should include a key selector procedure that retrieves the key associated with a given record. Now we represent the data base as a set of records. To locate the record with a given key we use a procedure lookup, which takes as arguments a key and a data base and which returns the record that has that key, or false if there is no such record. lookup is implemented in almost the same way as element-of-set?. For example, if the set of records is implemented as an unordered list, we could use (define (lookup given-key set-of-records) (cond ((null? set-of-records) false) ((equal? given-key (key (car set-of-records))) (car set-of-records)) (else (lookup given-key (cdr set-of-records))))) Of course, there are beer ways to represent large sets than as un- ordered lists. Information-retrieval systems in which records have to be “randomly accessed” are typically implemented by a tree-based method, such as the binary-tree representation discussed previously. In design- ing such a system the methodology of data abstraction can be a great help. e designer can create an initial implementation using a sim- ple, straightforward representation such as unordered lists. is will be unsuitable for the eventual system, but it can be useful in providing a “quick and dirty” data base with which to test the rest of the system. Later on, the data representation can be modiﬁed to be more sophisti- 217 cated. If the data base is accessed in terms of abstract selectors and con- structors, this change in representation will not require any changes to the rest of the system. Exercise 2.66: Implement the lookup procedure for the case where the set of records is structured as a binary tree, or- dered by the numerical values of the keys. 2.3.4 Example: Huﬀman Encoding Trees is section provides practice in the use of list structure and data ab- straction to manipulate sets and trees. e application is to methods for representing data as sequences of ones and zeros (bits). For example, the standard code used to represent text in computers encodes each character as a sequence of seven bits. Using seven bits allows us to distinguish 27 , or 128, possible diﬀerent characters. In general, if we want to distinguish n diﬀerent symbols, we will need to use log2 n bits per symbol. If all our messages are made up of the eight symbols A, B, C, D, E, F, G, and H, we can choose a code with three bits per character, for example A 000 C 010 E 100 G 110 B 001 D 011 F 101 H 111 With this code, the message BACADAEAFABBAAAGAH is encoded as the string of 54 bits 001000010000011000100000101000001001000000000110000111 218 Codes such as and the A-through-H code above are known as ﬁxed-length codes, because they represent each symbol in the message with the same number of bits. It is sometimes advantageous to use variable- length codes, in which diﬀerent symbols may be represented by diﬀer- ent numbers of bits. For example, Morse code does not use the same number of dots and dashes for each leer of the alphabet. In particular, E, the most frequent leer, is represented by a single dot. In general, if our messages are such that some symbols appear very frequently and some very rarely, we can encode data more eﬃciently (i.e., using fewer bits per message) if we assign shorter codes to the frequent symbols. Consider the following alternative code for the leers A through H: A 0 C 1010 E 1100 G 1110 B 100 D 1011 F 1101 H 1111 With this code, the same message as above is encoded as the string 100010100101101100011010100100000111001111 is string contains 42 bits, so it saves more than 20% in space in com- parison with the ﬁxed-length code shown above. One of the diﬃculties of using a variable-length code is knowing when you have reached the end of a symbol in reading a sequence of zeros and ones. Morse code solves this problem by using a special sep- arator code (in this case, a pause) aer the sequence of dots and dashes for each leer. Another solution is to design the code in such a way that no complete code for any symbol is the beginning (or preﬁx ) of the code for another symbol. Such a code is called a preﬁx code. In the example above, A is encoded by 0 and B is encoded by 100, so no other symbol can have a code that begins with 0 or with 100. 219 In general, we can aain signiﬁcant savings if we use variable-length preﬁx codes that take advantage of the relative frequencies of the sym- bols in the messages to be encoded. One particular scheme for doing this is called the Huﬀman encoding method, aer its discoverer, David Huﬀman. A Huﬀman code can be represented as a binary tree whose leaves are the symbols that are encoded. At each non-leaf node of the tree there is a set containing all the symbols in the leaves that lie below the node. In addition, each symbol at a leaf is assigned a weight (which is its relative frequency), and each non-leaf node contains a weight that is the sum of all the weights of the leaves lying below it. e weights are not used in the encoding or the decoding process. We will see below how they are used to help construct the tree. Figure 2.18 shows the Huﬀman tree for the A-through-H code given above. e weights at the leaves indicate that the tree was designed for messages in which A appears with relative frequency 8, B with relative frequency 3, and the other leers each with relative frequency 1. Given a Huﬀman tree, we can ﬁnd the encoding of any symbol by starting at the root and moving down until we reach the leaf that holds the symbol. Each time we move down a le branch we add a 0 to the code, and each time we move down a right branch we add a 1. (We decide which branch to follow by testing to see which branch either is the leaf node for the symbol or contains the symbol in its set.) For example, starting from the root of the tree in Figure 2.18, we arrive at the leaf for D by following a right branch, then a le branch, then a right branch, then a right branch; hence, the code for D is 1011. To decode a bit sequence using a Huﬀman tree, we begin at the root and use the successive zeros and ones of the bit sequence to determine whether to move down the le or the right branch. Each time we come to a leaf, we have generated a new symbol in the message, at which 220 {A B C D E F G H} 17 {B C D E F G H} 9 A 8 {E F G H} 4 {B C D} 5 {E F} 2 {C D} 2 {G H} 2 B 3 E 1 F 1 C 1 D 1 G 1 H 1 Figure 2.18: A Huﬀman encoding tree. point we start over from the root of the tree to ﬁnd the next symbol. For example, suppose we are given the tree above and the sequence 10001010. Starting at the root, we move down the right branch, (since the ﬁrst bit of the string is 1), then down the le branch (since the second bit is 0), then down the le branch (since the third bit is also 0). is brings us to the leaf for B, so the ﬁrst symbol of the decoded message is B. Now we start again at the root, and we make a le move because the next bit in the string is 0. is brings us to the leaf for A. en we start again at the root with the rest of the string 1010, so we move right, le, right, le and reach C. us, the entire message is BAC. Generating Huﬀman trees Given an “alphabet” of symbols and their relative frequencies, how do we construct the “best” code? (In other words, which tree will encode messages with the fewest bits?) Huﬀman gave an algorithm for doing 221 this and showed that the resulting code is indeed the best variable- length code for messages where the relative frequency of the symbols matches the frequencies with which the code was constructed. We will not prove this optimality of Huﬀman codes here, but we will show how Huﬀman trees are constructed.42 e algorithm for generating a Huﬀman tree is very simple. e idea is to arrange the tree so that the symbols with the lowest frequency appear farthest away from the root. Begin with the set of leaf nodes, containing symbols and their frequencies, as determined by the initial data from which the code is to be constructed. Now ﬁnd two leaves with the lowest weights and merge them to produce a node that has these two nodes as its le and right branches. e weight of the new node is the sum of the two weights. Remove the two leaves from the original set and replace them by this new node. Now continue this process. At each step, merge two nodes with the smallest weights, removing them from the set and replacing them with a node that has these two as its le and right branches. e process stops when there is only one node le, which is the root of the entire tree. Here is how the Huﬀman tree of Figure 2.18 was generated: Initial leaves {(A 8) (B 3) (C 1) (D 1) (E 1) (F 1) (G 1) (H 1)} Merge {(A 8) (B 3) ({C D} 2) (E 1) (F 1) (G 1) (H 1)} Merge {(A 8) (B 3) ({C D} 2) ({E F} 2) (G 1) (H 1)} Merge {(A 8) (B 3) ({C D} 2) ({E F} 2) ({G H} 2)} Merge {(A 8) (B 3) ({C D} 2) ({E F G H} 4)} Merge {(A 8) ({B C D} 5) ({E F G H} 4)} Merge {(A 8) ({B C D E F G H} 9)} Final merge {({A B C D E F G H} 17)} 42 See Hamming 1980 for a discussion of the mathematical properties of Huﬀman codes. 222 e algorithm does not always specify a unique tree, because there may not be unique smallest-weight nodes at each step. Also, the choice of the order in which the two nodes are merged (i.e., which will be the right branch and which will be the le branch) is arbitrary. Representing Huﬀman trees In the exercises below we will work with a system that uses Huﬀman trees to encode and decode messages and generates Huﬀman trees ac- cording to the algorithm outlined above. We will begin by discussing how trees are represented. Leaves of the tree are represented by a list consisting of the symbol leaf, the symbol at the leaf, and the weight: (define (make-leaf symbol weight) (list 'leaf symbol weight)) (define (leaf? object) (eq? (car object) 'leaf)) (define (symbol-leaf x) (cadr x)) (define (weight-leaf x) (caddr x)) A general tree will be a list of a le branch, a right branch, a set of symbols, and a weight. e set of symbols will be simply a list of the symbols, rather than some more sophisticated set representation. When we make a tree by merging two nodes, we obtain the weight of the tree as the sum of the weights of the nodes, and the set of symbols as the union of the sets of symbols for the nodes. Since our symbol sets are represented as lists, we can form the union by using the append procedure we deﬁned in Section 2.2.1: (define (make-code-tree left right) (list left right (append (symbols left) (symbols right)) (+ (weight left) (weight right)))) 223 If we make a tree in this way, we have the following selectors: (define (left-branch tree) (car tree)) (define (right-branch tree) (cadr tree)) (define (symbols tree) (if (leaf? tree) (list (symbol-leaf tree)) (caddr tree))) (define (weight tree) (if (leaf? tree) (weight-leaf tree) (cadddr tree))) e procedures symbols and weight must do something slightly diﬀer- ent depending on whether they are called with a leaf or a general tree. ese are simple examples of generic procedures (procedures that can handle more than one kind of data), which we will have much more to say about in Section 2.4 and Section 2.5. The decoding procedure e following procedure implements the decoding algorithm. It takes as arguments a list of zeros and ones, together with a Huﬀman tree. (define (decode bits tree) (define (decode-1 bits current-branch) (if (null? bits) '() (let ((next-branch (choose-branch (car bits) current-branch))) (if (leaf? next-branch) (cons (symbol-leaf next-branch) (decode-1 (cdr bits) tree)) (decode-1 (cdr bits) next-branch))))) (decode-1 bits tree)) 224 (define (choose-branch bit branch) (cond ((= bit 0) (left-branch branch)) ((= bit 1) (right-branch branch)) (else (error "bad bit: CHOOSE-BRANCH" bit)))) e procedure decode-1 takes two arguments: the list of remaining bits and the current position in the tree. It keeps moving “down” the tree, choosing a le or a right branch according to whether the next bit in the list is a zero or a one. (is is done with the procedure choose-branch.) When it reaches a leaf, it returns the symbol at that leaf as the next symbol in the message by consing it onto the result of decoding the rest of the message, starting at the root of the tree. Note the error check in the ﬁnal clause of choose-branch, which complains if the procedure ﬁnds something other than a zero or a one in the input data. Sets of weighted elements In our representation of trees, each non-leaf node contains a set of sym- bols, which we have represented as a simple list. However, the tree- generating algorithm discussed above requires that we also work with sets of leaves and trees, successively merging the two smallest items. Since we will be required to repeatedly ﬁnd the smallest item in a set, it is convenient to use an ordered representation for this kind of set. We will represent a set of leaves and trees as a list of elements, ar- ranged in increasing order of weight. e following adjoin-set pro- cedure for constructing sets is similar to the one described in Exercise 2.61; however, items are compared by their weights, and the element being added to the set is never already in it. (define (adjoin-set x set) (cond ((null? set) (list x)) ((< (weight x) (weight (car set))) (cons x set)) 225 (else (cons (car set) (adjoin-set x (cdr set)))))) e following procedure takes a list of symbol-frequency pairs such as ((A 4) (B 2) (C 1) (D 1)) and constructs an initial ordered set of leaves, ready to be merged according to the Huﬀman algorithm: (define (make-leaf-set pairs) (if (null? pairs) '() (let ((pair (car pairs))) (adjoin-set (make-leaf (car pair) ; symbol (cadr pair)) ; frequency (make-leaf-set (cdr pairs)))))) Exercise 2.67: Deﬁne an encoding tree and a sample mes- sage: (define sample-tree (make-code-tree (make-leaf 'A 4) (make-code-tree (make-leaf 'B 2) (make-code-tree (make-leaf 'D 1) (make-leaf 'C 1))))) (define sample-message '(0 1 1 0 0 1 0 1 0 1 1 1 0)) Use the decode procedure to decode the message, and give the result. Exercise 2.68: e encode procedure takes as arguments a message and a tree and produces the list of bits that gives the encoded message. 226 (define (encode message tree) (if (null? message) '() (append (encode-symbol (car message) tree) (encode (cdr message) tree)))) encode-symbol is a procedure, which you must write, that returns the list of bits that encodes a given symbol accord- ing to a given tree. You should design encode-symbol so that it signals an error if the symbol is not in the tree at all. Test your procedure by encoding the result you obtained in Exercise 2.67 with the sample tree and seeing whether it is the same as the original sample message. Exercise 2.69: e following procedure takes as its argu- ment a list of symbol-frequency pairs (where no symbol appears in more than one pair) and generates a Huﬀman encoding tree according to the Huﬀman algorithm. (define (generate-huffman-tree pairs) (successive-merge (make-leaf-set pairs))) make-leaf-set is the procedure given above that trans- forms the list of pairs into an ordered set of leaves. successive- merge is the procedure you must write, using make-code- tree to successively merge the smallest-weight elements of the set until there is only one element le, which is the desired Huﬀman tree. (is procedure is slightly tricky, but not really complicated. If you ﬁnd yourself designing a com- plex procedure, then you are almost certainly doing some- thing wrong. You can take signiﬁcant advantage of the fact that we are using an ordered set representation.) 227 Exercise 2.70: e following eight-symbol alphabet with associated relative frequencies was designed to eﬃciently encode the lyrics of 1950s rock songs. (Note that the “sym- bols” of an “alphabet” need not be individual leers.) A 2 GET 2 SHA 3 WAH 1 BOOM 1 JOB 2 NA 16 YIP 9 Use generate-huffman-tree (Exercise 2.69) to generate a corresponding Huﬀman tree, and use encode (Exercise 2.68) to encode the following message: Get a job Sha na na na na na na na na Get a job Sha na na na na na na na na Wah yip yip yip yip yip yip yip yip yip Sha boom How many bits are required for the encoding? What is the smallest number of bits that would be needed to encode this song if we used a ﬁxed-length code for the eight-symbol alphabet? Exercise 2.71: Suppose we have a Huﬀman tree for an al- phabet of n symbols, and that the relative frequencies of the symbols are 1, 2, 4, . . . , 2n −1 . Sketch the tree for n = 5; for n = 10. In such a tree (for general n) how many bits are required to encode the most frequent symbol? e least frequent symbol? 228 Exercise 2.72: Consider the encoding procedure that you designed in Exercise 2.68. What is the order of growth in the number of steps needed to encode a symbol? Be sure to include the number of steps needed to search the sym- bol list at each node encountered. To answer this question in general is diﬃcult. Consider the special case where the relative frequencies of the n symbols are as described in Ex- ercise 2.71, and give the order of growth (as a function of n) of the number of steps needed to encode the most frequent and least frequent symbols in the alphabet. 2.4 Multiple Representations for Abstract Data We have introduced data abstraction, a methodology for structuring systems in such a way that much of a program can be speciﬁed indepen- dent of the choices involved in implementing the data objects that the program manipulates. For example, we saw in Section 2.1.1 how to sep- arate the task of designing a program that uses rational numbers from the task of implementing rational numbers in terms of the computer language’s primitive mechanisms for constructing compound data. e key idea was to erect an abstraction barrier—in this case, the selec- tors and constructors for rational numbers (make-rat, numer, denom)— that isolates the way rational numbers are used from their underlying representation in terms of list structure. A similar abstraction barrier isolates the details of the procedures that perform rational arithmetic (add-rat, sub-rat, mul-rat, and div-rat) from the “higher-level” pro- cedures that use rational numbers. e resulting program has the struc- ture shown in Figure 2.1. ese data-abstraction barriers are powerful tools for controlling 229 complexity. By isolating the underlying representations of data objects, we can divide the task of designing a large program into smaller tasks that can be performed separately. But this kind of data abstraction is not yet powerful enough, because it may not always make sense to speak of “the underlying representation” for a data object. For one thing, there might be more than one useful representation for a data object, and we might like to design systems that can deal with multiple representations. To take a simple example, complex numbers may be represented in two almost equivalent ways: in rectangular form (real and imaginary parts) and in polar form (magnitude and angle). Sometimes rectangular form is more appropriate and sometimes polar form is more appropriate. Indeed, it is perfectly plausible to imagine a system in which complex numbers are represented in both ways, and in which the procedures for manipulating complex numbers work with either representation. More importantly, programming systems are oen designed by many people working over extended periods of time, subject to requirements that change over time. In such an environment, it is simply not possi- ble for everyone to agree in advance on choices of data representation. So in addition to the data-abstraction barriers that isolate representa- tion from use, we need abstraction barriers that isolate diﬀerent de- sign choices from each other and permit diﬀerent choices to coexist in a single program. Furthermore, since large programs are oen created by combining pre-existing modules that were designed in isolation, we need conventions that permit programmers to incorporate modules into larger systems additively, that is, without having to redesign or reimple- ment these modules. In this section, we will learn how to cope with data that may be represented in diﬀerent ways by diﬀerent parts of a program. is re- 230 quires constructing generic procedures—procedures that can operate on data that may be represented in more than one way. Our main technique for building generic procedures will be to work in terms of data objects that have type tags, that is, data objects that include explicit information about how they are to be processed. We will also discuss data-directed programming, a powerful and convenient implementation strategy for additively assembling systems with generic operations. We begin with the simple complex-number example. We will see how type tags and data-directed style enable us to design separate rect- angular and polar representations for complex numbers while main- taining the notion of an abstract “complex-number” data object. We will accomplish this by deﬁning arithmetic procedures for complex numbers (add-complex, sub-complex, mul-complex, and div-complex) in terms of generic selectors that access parts of a complex number independent of how the number is represented. e resulting complex-number sys- tem, as shown in Figure 2.19, contains two diﬀerent kinds of abstrac- tion barriers. e “horizontal” abstraction barriers play the same role as the ones in Figure 2.1. ey isolate “higher-level” operations from “lower-level” representations. In addition, there is a “vertical” barrier that gives us the ability to separately design and install alternative rep- resentations. In Section 2.5 we will show how to use type tags and data-directed style to develop a generic arithmetic package. is provides procedures (add, mul, and so on) that can be used to manipulate all sorts of “num- bers” and can be easily extended when a new kind of number is needed. In Section 2.5.3, we’ll show how to use generic arithmetic in a system that performs symbolic algebra. 231 Programs that use complex numbers add-complex sub-complex mul-complex div-complex Complex-arithmetic package Rectangular Polar representation representation List structure and primitive machine arithmetic Figure 2.19: Data-abstraction barriers in the complex- number system. 2.4.1 Representations for Complex Numbers We will develop a system that performs arithmetic operations on com- plex numbers as a simple but unrealistic example of a program that uses generic operations. We begin by discussing two plausible representa- tions for complex numbers as ordered pairs: rectangular form (real part and imaginary part) and polar form (magnitude and angle).43 Section 2.4.2 will show how both representations can be made to coexist in a single system through the use of type tags and generic operations. Like rational numbers, complex numbers are naturally represented as ordered pairs. e set of complex numbers can be thought of as a two-dimensional space with two orthogonal axes, the “real” axis and the 43 In actual computational systems, rectangular form is preferable to polar form most of the time because of roundoﬀ errors in conversion between rectangular and polar form. is is why the complex-number example is unrealistic. Nevertheless, it provides a clear illustration of the design of a system using generic operations and a good intro- duction to the more substantial systems to be developed later in this chapter. 232 Imaginary y z = x + iy = re iA r A Real x Figure 2.20: Complex numbers as points in the plane. “imaginary” axis. (See Figure 2.20.) From this point of view, the complex number z = x + iy (where i 2 = −1) can be thought of as the point in the plane whose real coordinate is x and whose imaginary coordinate is y. Addition of complex numbers reduces in this representation to addition of coordinates: Real-part(z 1 + z 2 ) = Real-part(z 1 ) + Real-part(z 2 ), Imaginary-part(z 1 + z 2 ) = Imaginary-part(z 1 ) + Imaginary-part(z 2 ). When multiplying complex numbers, it is more natural to think in terms of representing a complex number in polar form, as a magnitude and an angle (r and A in Figure 2.20). e product of two complex num- bers is the vector obtained by stretching one complex number by the length of the other and then rotating it through the angle of the other: Magnitude(z 1 · z 2 ) = Magnitude(z 1 ) · Magnitude(z 2 ), Angle(z 1 · z 2 ) = Angle(z 1 ) + Angle(z 2 ). us, there are two diﬀerent representations for complex numbers, which 233 are appropriate for diﬀerent operations. Yet, from the viewpoint of some- one writing a program that uses complex numbers, the principle of data abstraction suggests that all the operations for manipulating complex numbers should be available regardless of which representation is used by the computer. For example, it is oen useful to be able to ﬁnd the magnitude of a complex number that is speciﬁed by rectangular coor- dinates. Similarly, it is oen useful to be able to determine the real part of a complex number that is speciﬁed by polar coordinates. To design such a system, we can follow the same data-abstraction strategy we followed in designing the rational-number package in Sec- tion 2.1.1. Assume that the operations on complex numbers are imple- mented in terms of four selectors: real-part, imag-part, magnitude and angle. Also assume that we have two procedures for construct- ing complex numbers: make-from-real-imag returns a complex num- ber with speciﬁed real and imaginary parts, and make-from-mag-ang returns a complex number with speciﬁed magnitude and angle. ese procedures have the property that, for any complex number z, both (make-from-real-imag (real-part z) (imag-part z)) and (make-from-mag-ang (magnitude z) (angle z)) produce complex numbers that are equal to z. Using these constructors and selectors, we can implement arith- metic on complex numbers using the “abstract data” speciﬁed by the constructors and selectors, just as we did for rational numbers in Sec- tion 2.1.1. As shown in the formulas above, we can add and subtract complex numbers in terms of real and imaginary parts while multiply- ing and dividing complex numbers in terms of magnitudes and angles: 234 (define (add-complex z1 z2) (make-from-real-imag (+ (real-part z1) (real-part z2)) (+ (imag-part z1) (imag-part z2)))) (define (sub-complex z1 z2) (make-from-real-imag (- (real-part z1) (real-part z2)) (- (imag-part z1) (imag-part z2)))) (define (mul-complex z1 z2) (make-from-mag-ang (* (magnitude z1) (magnitude z2)) (+ (angle z1) (angle z2)))) (define (div-complex z1 z2) (make-from-mag-ang (/ (magnitude z1) (magnitude z2)) (- (angle z1) (angle z2)))) To complete the complex-number package, we must choose a represen- tation and we must implement the constructors and selectors in terms of primitive numbers and primitive list structure. ere are two obvi- ous ways to do this: We can represent a complex number in “rectangular form” as a pair (real part, imaginary part) or in “polar form” as a pair (magnitude, angle). Which shall we choose? In order to make the diﬀerent choices concrete, imagine that there are two programmers, Ben Bitdiddle and Alyssa P. Hacker, who are independently designing representations for the complex-number sys- tem. Ben chooses to represent complex numbers in rectangular form. With this choice, selecting the real and imaginary parts of a complex number is straightforward, as is constructing a complex number with given real and imaginary parts. To ﬁnd the magnitude and the angle, or to construct a complex number with a given magnitude and angle, he uses the trigonometric relations √ x = r cos A, r = x 2 + y2 , y = r sin A, A = arctan(y, x), which relate the real and imaginary parts (x , y) to the magnitude and the 235 angle (r, A).44 Ben’s representation is therefore given by the following selectors and constructors: (define (real-part z) (car z)) (define (imag-part z) (cdr z)) (define (magnitude z) (sqrt (+ (square (real-part z)) (square (imag-part z))))) (define (angle z) (atan (imag-part z) (real-part z))) (define (make-from-real-imag x y) (cons x y)) (define (make-from-mag-ang r a) (cons (* r (cos a)) (* r (sin a)))) Alyssa, in contrast, chooses to represent complex numbers in polar form. For her, selecting the magnitude and angle is straightforward, but she has to use the trigonometric relations to obtain the real and imaginary parts. Alyssa’s representation is: (define (real-part z) (* (magnitude z) (cos (angle z)))) (define (imag-part z) (* (magnitude z) (sin (angle z)))) (define (magnitude z) (car z)) (define (angle z) (cdr z)) (define (make-from-real-imag x y) (cons (sqrt (+ (square x) (square y))) (atan y x))) (define (make-from-mag-ang r a) (cons r a)) e discipline of data abstraction ensures that the same implementation of add-complex, sub-complex, mul-complex, and div-complex will work with either Ben’s representation or Alyssa’s representation. 44 e arctangent function referred to here, computed by Scheme’s atan procedure, is deﬁned so as to take two arguments y and x and to return the angle whose tangent is y/x . e signs of the arguments determine the quadrant of the angle. 236 2.4.2 Tagged data One way to view data abstraction is as an application of the “princi- ple of least commitment.” In implementing the complex-number system in Section 2.4.1, we can use either Ben’s rectangular representation or Alyssa’s polar representation. e abstraction barrier formed by the se- lectors and constructors permits us to defer to the last possible moment the choice of a concrete representation for our data objects and thus retain maximum ﬂexibility in our system design. e principle of least commitment can be carried to even further extremes. If we desire, we can maintain the ambiguity of representation even aer we have designed the selectors and constructors, and elect to use both Ben’s representation and Alyssa’s representation. If both representations are included in a single system, however, we will need some way to distinguish data in polar form from data in rectangular form. Otherwise, if we were asked, for instance, to ﬁnd the magnitude of the pair (3, 4), we wouldn’t know whether to answer 5 (interpreting the number in rectangular form) or 3 (interpreting the number in polar form). A straightforward way to accomplish this distinction is to include a type tag —the symbol rectangular or polar—as part of each complex number. en when we need to manipulate a complex number we can use the tag to decide which selector to apply. In order to manipulate tagged data, we will assume that we have procedures type-tag and contents that extract from a data object the tag and the actual contents (the polar or rectangular coordinates, in the case of a complex number). We will also postulate a procedure attach- tag that takes a tag and contents and produces a tagged data object. A straightforward way to implement this is to use ordinary list structure: (define (attach-tag type-tag contents) (cons type-tag contents)) 237 (define (type-tag datum) (if (pair? datum) (car datum) (error "Bad tagged datum: TYPE-TAG" datum))) (define (contents datum) (if (pair? datum) (cdr datum) (error "Bad tagged datum: CONTENTS" datum))) Using these procedures, we can deﬁne predicates rectangular? and polar?, which recognize rectangular and polar numbers, respectively: (define (rectangular? z) (eq? (type-tag z) 'rectangular)) (define (polar? z) (eq? (type-tag z) 'polar)) With type tags, Ben and Alyssa can now modify their code so that their two diﬀerent representations can coexist in the same system. When- ever Ben constructs a complex number, he tags it as rectangular. When- ever Alyssa constructs a complex number, she tags it as polar. In addi- tion, Ben and Alyssa must make sure that the names of their proce- dures do not conﬂict. One way to do this is for Ben to append the suﬃx rectangular to the name of each of his representation procedures and for Alyssa to append polar to the names of hers. Here is Ben’s revised rectangular representation from Section 2.4.1: (define (real-part-rectangular z) (car z)) (define (imag-part-rectangular z) (cdr z)) (define (magnitude-rectangular z) (sqrt (+ (square (real-part-rectangular z)) (square (imag-part-rectangular z))))) (define (angle-rectangular z) (atan (imag-part-rectangular z) (real-part-rectangular z))) 238 (define (make-from-real-imag-rectangular x y) (attach-tag 'rectangular (cons x y))) (define (make-from-mag-ang-rectangular r a) (attach-tag 'rectangular (cons (* r (cos a)) (* r (sin a))))) and here is Alyssa’s revised polar representation: (define (real-part-polar z) (* (magnitude-polar z) (cos (angle-polar z)))) (define (imag-part-polar z) (* (magnitude-polar z) (sin (angle-polar z)))) (define (magnitude-polar z) (car z)) (define (angle-polar z) (cdr z)) (define (make-from-real-imag-polar x y) (attach-tag 'polar (cons (sqrt (+ (square x) (square y))) (atan y x)))) (define (make-from-mag-ang-polar r a) (attach-tag 'polar (cons r a))) Each generic selector is implemented as a procedure that checks the tag of its argument and calls the appropriate procedure for handling data of that type. For example, to obtain the real part of a complex number, real-part examines the tag to determine whether to use Ben’s real- part-rectangular or Alyssa’s real-part-polar. In either case, we use contents to extract the bare, untagged datum and send this to the rect- angular or polar procedure as required: (define (real-part z) (cond ((rectangular? z) (real-part-rectangular (contents z))) ((polar? z) (real-part-polar (contents z))) (else (error "Unknown type: REAL-PART" z)))) 239 (define (imag-part z) (cond ((rectangular? z) (imag-part-rectangular (contents z))) ((polar? z) (imag-part-polar (contents z))) (else (error "Unknown type: IMAG-PART" z)))) (define (magnitude z) (cond ((rectangular? z) (magnitude-rectangular (contents z))) ((polar? z) (magnitude-polar (contents z))) (else (error "Unknown type: MAGNITUDE" z)))) (define (angle z) (cond ((rectangular? z) (angle-rectangular (contents z))) ((polar? z) (angle-polar (contents z))) (else (error "Unknown type: ANGLE" z)))) To implement the complex-number arithmetic operations, we can use the same procedures add-complex, sub-complex, mul-complex, and div- complex from Section 2.4.1, because the selectors they call are generic, and so will work with either representation. For example, the procedure add-complex is still (define (add-complex z1 z2) (make-from-real-imag (+ (real-part z1) (real-part z2)) (+ (imag-part z1) (imag-part z2)))) Finally, we must choose whether to construct complex numbers using Ben’s representation or Alyssa’s representation. One reasonable choice is to construct rectangular numbers whenever we have real and imag- inary parts and to construct polar numbers whenever we have magni- tudes and angles: 240 Programs that use complex numbers add-complex sub-complex mul-complex div-complex Complex-arithmetic package real-part magnitude imag-part angle Rectangular Polar representation representation List structure and primitive machine arithmetic Figure 2.21: Structure of the generic complex-arithmetic system. (define (make-from-real-imag x y) (make-from-real-imag-rectangular x y)) (define (make-from-mag-ang r a) (make-from-mag-ang-polar r a)) e resulting complex-number system has the structure shown in Fig- ure 2.21. e system has been decomposed into three relatively inde- pendent parts: the complex-number-arithmetic operations, Alyssa’s po- lar implementation, and Ben’s rectangular implementation. e polar and rectangular implementations could have been wrien by Ben and Alyssa working separately, and both of these can be used as underly- ing representations by a third programmer implementing the complex- arithmetic procedures in terms of the abstract constructor/selector in- terface. Since each data object is tagged with its type, the selectors operate on the data in a generic manner. at is, each selector is deﬁned to have a behavior that depends upon the particular type of data it is applied to. 241 Notice the general mechanism for interfacing the separate representa- tions: Within a given representation implementation (say, Alyssa’s po- lar package) a complex number is an untyped pair (magnitude, angle). When a generic selector operates on a number of polar type, it strips oﬀ the tag and passes the contents on to Alyssa’s code. Conversely, when Alyssa constructs a number for general use, she tags it with a type so that it can be appropriately recognized by the higher-level procedures. is discipline of stripping oﬀ and aaching tags as data objects are passed from level to level can be an important organizational strategy, as we shall see in Section 2.5. 2.4.3 Data-Directed Programming and Additivity e general strategy of checking the type of a datum and calling an appropriate procedure is called dispatching on type. is is a powerful strategy for obtaining modularity in system design. On the other hand, implementing the dispatch as in Section 2.4.2 has two signiﬁcant weak- nesses. One weakness is that the generic interface procedures (real- part, imag-part, magnitude, and angle) must know about all the dif- ferent representations. For instance, suppose we wanted to incorporate a new representation for complex numbers into our complex-number system. We would need to identify this new representation with a type, and then add a clause to each of the generic interface procedures to check for the new type and apply the appropriate selector for that rep- resentation. Another weakness of the technique is that even though the indi- vidual representations can be designed separately, we must guarantee that no two procedures in the entire system have the same name. is is why Ben and Alyssa had to change the names of their original proce- dures from Section 2.4.1. 242 e issue underlying both of these weaknesses is that the technique for implementing generic interfaces is not additive. e person imple- menting the generic selector procedures must modify those procedures each time a new representation is installed, and the people interfacing the individual representations must modify their code to avoid name conﬂicts. In each of these cases, the changes that must be made to the code are straightforward, but they must be made nonetheless, and this is a source of inconvenience and error. is is not much of a problem for the complex-number system as it stands, but suppose there were not two but hundreds of diﬀerent representations for complex numbers. And suppose that there were many generic selectors to be maintained in the abstract-data interface. Suppose, in fact, that no one program- mer knew all the interface procedures or all the representations. e problem is real and must be addressed in such programs as large-scale data-base-management systems. What we need is a means for modularizing the system design even further. is is provided by the programming technique known as data- directed programming. To understand how data-directed programming works, begin with the observation that whenever we deal with a set of generic operations that are common to a set of diﬀerent types we are, in eﬀect, dealing with a two-dimensional table that contains the possi- ble operations on one axis and the possible types on the other axis. e entries in the table are the procedures that implement each operation for each type of argument presented. In the complex-number system developed in the previous section, the correspondence between opera- tion name, data type, and actual procedure was spread out among the various conditional clauses in the generic interface procedures. But the same information could have been organized in a table, as shown in Figure 2.22. 243 Types Polar Rectangular real-part real-part-polar real-part-rectangular Operations imag-part imag-part-polar imag-part-rectangular magnitude magnitude-polar magnitude-rectangular angle angle-polar angle-rectangular Figure 2.22: Table of operations for the complex-number system. Data-directed programming is the technique of designing programs to work with such a table directly. Previously, we implemented the mechanism that interfaces the complex-arithmetic code with the two representation packages as a set of procedures that each perform an explicit dispatch on type. Here we will implement the interface as a sin- gle procedure that looks up the combination of the operation name and argument type in the table to ﬁnd the correct procedure to apply, and then applies it to the contents of the argument. If we do this, then to add a new representation package to the system we need not change any existing procedures; we need only add new entries to the table. To implement this plan, assume that we have two procedures, put and get, for manipulating the operation-and-type table: • (put ⟨op ⟩ ⟨type ⟩ ⟨item ⟩) installs the ⟨item ⟩ in the table, indexed by the ⟨op ⟩ and the ⟨type ⟩. • (get ⟨op ⟩ ⟨type ⟩) looks up the ⟨op ⟩, ⟨type ⟩ entry in the table and returns the item found there. If no item is found, get returns false. For now, we can assume that put and get are included in our language. In Chapter 3 (Section 3.3.3) we will see how to implement these and 244 other operations for manipulating tables. Here is how data-directed programming can be used in the complex- number system. Ben, who developed the rectangular representation, implements his code just as he did originally. He deﬁnes a collection of procedures, or a package, and interfaces these to the rest of the sys- tem by adding entries to the table that tell the system how to operate on rectangular numbers. is is accomplished by calling the following procedure: (define (install-rectangular-package) ;; internal procedures (define (real-part z) (car z)) (define (imag-part z) (cdr z)) (define (make-from-real-imag x y) (cons x y)) (define (magnitude z) (sqrt (+ (square (real-part z)) (square (imag-part z))))) (define (angle z) (atan (imag-part z) (real-part z))) (define (make-from-mag-ang r a) (cons (* r (cos a)) (* r (sin a)))) ;; interface to the rest of the system (define (tag x) (attach-tag 'rectangular x)) (put 'real-part '(rectangular) real-part) (put 'imag-part '(rectangular) imag-part) (put 'magnitude '(rectangular) magnitude) (put 'angle '(rectangular) angle) (put 'make-from-real-imag 'rectangular (lambda (x y) (tag (make-from-real-imag x y)))) (put 'make-from-mag-ang 'rectangular (lambda (r a) (tag (make-from-mag-ang r a)))) 'done) 245 Notice that the internal procedures here are the same procedures from Section 2.4.1 that Ben wrote when he was working in isolation. No changes are necessary in order to interface them to the rest of the sys- tem. Moreover, since these procedure deﬁnitions are internal to the in- stallation procedure, Ben needn’t worry about name conﬂicts with other procedures outside the rectangular package. To interface these to the rest of the system, Ben installs his real-part procedure under the op- eration name real-part and the type (rectangular), and similarly for the other selectors.45 e interface also deﬁnes the constructors to be used by the external system.46 ese are identical to Ben’s internally deﬁned constructors, except that they aach the tag. Alyssa’s polar package is analogous: (define (install-polar-package) ;; internal procedures (define (magnitude z) (car z)) (define (angle z) (cdr z)) (define (make-from-mag-ang r a) (cons r a)) (define (real-part z) (* (magnitude z) (cos (angle z)))) (define (imag-part z) (* (magnitude z) (sin (angle z)))) (define (make-from-real-imag x y) (cons (sqrt (+ (square x) (square y))) (atan y x))) ;; interface to the rest of the system (define (tag x) (attach-tag 'polar x)) (put 'real-part '(polar) real-part) (put 'imag-part '(polar) imag-part) (put 'magnitude '(polar) magnitude) 45 We use the list (rectangular) rather than the symbol rectangular to allow for the possibility of operations with multiple arguments, not all of the same type. 46 e type the constructors are installed under needn’t be a list because a constructor is always used to make an object of one particular type. 246 (put 'angle '(polar) angle) (put 'make-from-real-imag 'polar (lambda (x y) (tag (make-from-real-imag x y)))) (put 'make-from-mag-ang 'polar (lambda (r a) (tag (make-from-mag-ang r a)))) 'done) Even though Ben and Alyssa both still use their original procedures deﬁned with the same names as each other’s (e.g., real-part), these deﬁnitions are now internal to diﬀerent procedures (see Section 1.1.8), so there is no name conﬂict. e complex-arithmetic selectors access the table by means of a general “operation” procedure called apply-generic, which applies a generic operation to some arguments. apply-generic looks in the ta- ble under the name of the operation and the types of the arguments and applies the resulting procedure if one is present:47 (define (apply-generic op . args) (let ((type-tags (map type-tag args))) (let ((proc (get op type-tags))) (if proc (apply proc (map contents args)) (error 47 apply-generic uses the doed-tail notation described in Exercise 2.20, because dif- ferent generic operations may take diﬀerent numbers of arguments. In apply-generic, op has as its value the ﬁrst argument to apply-generic and args has as its value a list of the remaining arguments. apply-generic also uses the primitive procedure apply, which takes two arguments, a procedure and a list. apply applies the procedure, using the elements in the list as arguments. For example, (apply + (list 1 2 3 4)) returns 10. 247 "No method for these types: APPLY-GENERIC" (list op type-tags)))))) Using apply-generic, we can deﬁne our generic selectors as follows: (define (real-part z) (apply-generic 'real-part z)) (define (imag-part z) (apply-generic 'imag-part z)) (define (magnitude z) (apply-generic 'magnitude z)) (define (angle z) (apply-generic 'angle z)) Observe that these do not change at all if a new representation is added to the system. We can also extract from the table the constructors to be used by the programs external to the packages in making complex numbers from real and imaginary parts and from magnitudes and angles. As in Section 2.4.2, we construct rectangular numbers whenever we have real and imaginary parts, and polar numbers whenever we have magnitudes and angles: (define (make-from-real-imag x y) ((get 'make-from-real-imag 'rectangular) x y)) (define (make-from-mag-ang r a) ((get 'make-from-mag-ang 'polar) r a)) Exercise 2.73: Section 2.3.2 described a program that per- forms symbolic diﬀerentiation: (define (deriv exp var) (cond ((number? exp) 0) ((variable? exp) (if (same-variable? exp var) 1 0)) ((sum? exp) (make-sum (deriv (addend exp) var) (deriv (augend exp) var))) 248 ((product? exp) (make-sum (make-product (multiplier exp) (deriv (multiplicand exp) var)) (make-product (deriv (multiplier exp) var) (multiplicand exp)))) ⟨more rules can be added here⟩ (else (error "unknown expression type: DERIV" exp)))) We can regard this program as performing a dispatch on the type of the expression to be diﬀerentiated. In this situ- ation the “type tag” of the datum is the algebraic operator symbol (such as +) and the operation being performed is deriv. We can transform this program into data-directed style by rewriting the basic derivative procedure as (define (deriv exp var) (cond ((number? exp) 0) ((variable? exp) (if (same-variable? exp var) 1 0)) (else ((get 'deriv (operator exp)) (operands exp) var)))) (define (operator exp) (car exp)) (define (operands exp) (cdr exp)) a. Explain what was done above. Why can’t we assim- ilate the predicates number? and variable? into the data-directed dispatch? b. Write the procedures for derivatives of sums and prod- ucts, and the auxiliary code required to install them in the table used by the program above. 249 c. Choose any additional diﬀerentiation rule that you like, such as the one for exponents (Exercise 2.56), and install it in this data-directed system. d. In this simple algebraic manipulator the type of an expression is the algebraic operator that binds it to- gether. Suppose, however, we indexed the procedures in the opposite way, so that the dispatch line in deriv looked like ((get (operator exp) 'deriv) (operands exp) var) What corresponding changes to the derivative system are required? Exercise 2.74: Insatiable Enterprises, Inc., is a highly de- centralized conglomerate company consisting of a large num- ber of independent divisions located all over the world. e company’s computer facilities have just been interconnected by means of a clever network-interfacing scheme that makes the entire network appear to any user to be a single com- puter. Insatiable’s president, in her ﬁrst aempt to exploit the ability of the network to extract administrative infor- mation from division ﬁles, is dismayed to discover that, al- though all the division ﬁles have been implemented as data structures in Scheme, the particular data structure used varies from division to division. A meeting of division managers is hastily called to search for a strategy to integrate the ﬁles that will satisfy headquarters’ needs while preserving the existing autonomy of the divisions. Show how such a strategy can be implemented with data- directed programming. As an example, suppose that each 250 division’s personnel records consist of a single ﬁle, which contains a set of records keyed on employees’ names. e structure of the set varies from division to division. Fur- thermore, each employee’s record is itself a set (structured diﬀerently from division to division) that contains informa- tion keyed under identiﬁers such as address and salary. In particular: a. Implement for headquarters a get-record procedure that retrieves a speciﬁed employee’s record from a speciﬁed personnel ﬁle. e procedure should be ap- plicable to any division’s ﬁle. Explain how the individ- ual divisions’ ﬁles should be structured. In particular, what type information must be supplied? b. Implement for headquarters a get-salary procedure that returns the salary information from a given em- ployee’s record from any division’s personnel ﬁle. How should the record be structured in order to make this operation work? c. Implement for headquarters a find-employee-record procedure. is should search all the divisions’ ﬁles for the record of a given employee and return the record. Assume that this procedure takes as arguments an employee’s name and a list of all the divisions’ ﬁles. d. When Insatiable takes over a new company, what changes must be made in order to incorporate the new person- nel information into the central system? 251 Message passing e key idea of data-directed programming is to handle generic opera- tions in programs by dealing explicitly with operation-and-type tables, such as the table in Figure 2.22. e style of programming we used in Section 2.4.2 organized the required dispatching on type by having each operation take care of its own dispatching. In eﬀect, this decomposes the operation-and-type table into rows, with each generic operation proce- dure representing a row of the table. An alternative implementation strategy is to decompose the table into columns and, instead of using “intelligent operations” that dispatch on data types, to work with “intelligent data objects” that dispatch on operation names. We can do this by arranging things so that a data object, such as a rectangular number, is represented as a procedure that takes as input the required operation name and performs the operation indicated. In such a discipline, make-from-real-imag could be wrien as (define (make-from-real-imag x y) (define (dispatch op) (cond ((eq? op 'real-part) x) ((eq? op 'imag-part) y) ((eq? op 'magnitude) (sqrt (+ (square x) (square y)))) ((eq? op 'angle) (atan y x)) (else (error "Unknown op: MAKE-FROM-REAL-IMAG" op)))) dispatch) e corresponding apply-generic procedure, which applies a generic operation to an argument, now simply feeds the operation’s name to the data object and lets the object do the work:48 48 Onelimitation of this organization is it permits only generic procedures of one argument. 252 (define (apply-generic op arg) (arg op)) Note that the value returned by make-from-real-imag is a procedure— the internal dispatch procedure. is is the procedure that is invoked when apply-generic requests an operation to be performed. is style of programming is called message passing. e name comes from the image that a data object is an entity that receives the requested operation name as a “message.” We have already seen an example of message passing in Section 2.1.3, where we saw how cons, car, and cdr could be deﬁned with no data objects but only procedures. Here we see that message passing is not a mathematical trick but a useful technique for organizing systems with generic operations. In the remainder of this chapter we will continue to use data-directed programming, rather than message passing, to discuss generic arithmetic operations. In Chapter 3 we will return to message passing, and we will see that it can be a pow- erful tool for structuring simulation programs. Exercise 2.75: Implement the constructor make-from-mag- ang in message-passing style. is procedure should be anal- ogous to the make-from-real-imag procedure given above. Exercise 2.76: As a large system with generic operations evolves, new types of data objects or new operations may be needed. For each of the three strategies—generic opera- tions with explicit dispatch, data-directed style, and message- passing-style—describe the changes that must be made to a system in order to add new types or new operations. Which organization would be most appropriate for a system in which new types must oen be added? Which would be most appropriate for a system in which new operations must oen be added? 253 2.5 Systems with Generic Operations In the previous section, we saw how to design systems in which data objects can be represented in more than one way. e key idea is to link the code that speciﬁes the data operations to the several represen- tations by means of generic interface procedures. Now we will see how to use this same idea not only to deﬁne operations that are generic over diﬀerent representations but also to deﬁne operations that are generic over diﬀerent kinds of arguments. We have already seen several dif- ferent packages of arithmetic operations: the primitive arithmetic (+, -, *, /) built into our language, the rational-number arithmetic (add-rat, sub-rat, mul-rat, div-rat) of Section 2.1.1, and the complex-number arithmetic that we implemented in Section 2.4.3. We will now use data- directed techniques to construct a package of arithmetic operations that incorporates all the arithmetic packages we have already constructed. Figure 2.23 shows the structure of the system we shall build. Notice the abstraction barriers. From the perspective of someone using “num- bers,” there is a single procedure add that operates on whatever num- bers are supplied. add is part of a generic interface that allows the sep- arate ordinary-arithmetic, rational-arithmetic, and complex-arithmetic packages to be accessed uniformly by programs that use numbers. Any individual arithmetic package (such as the complex package) may it- self be accessed through generic procedures (such as add-complex) that combine packages designed for diﬀerent representations (such as rect- angular and polar). Moreover, the structure of the system is additive, so that one can design the individual arithmetic packages separately and combine them to produce a generic arithmetic system. 254 Programs that use numbers add sub mul div Generic arithmetic package add-rat sub-rat add-complex sub-complex + -- * / mul-rat div-rat mul-complex div-complex Complex arithmetic Rational Ordinary arithmetic arithmetic Rectangular Polar List structure and primitive machine arithmetic Figure 2.23: Generic arithmetic system. 2.5.1 Generic Arithmetic Operations e task of designing generic arithmetic operations is analogous to that of designing the generic complex-number operations. We would like, for instance, to have a generic addition procedure add that acts like or- dinary primitive addition + on ordinary numbers, like add-rat on ra- tional numbers, and like add-complex on complex numbers. We can implement add, and the other generic arithmetic operations, by follow- ing the same strategy we used in Section 2.4.3 to implement the generic selectors for complex numbers. We will aach a type tag to each kind of number and cause the generic procedure to dispatch to an appropriate package according to the data type of its arguments. e generic arithmetic procedures are deﬁned as follows: (define (add x y) (apply-generic 'add x y)) 255 (define (sub x y) (apply-generic 'sub x y)) (define (mul x y) (apply-generic 'mul x y)) (define (div x y) (apply-generic 'div x y)) We begin by installing a package for handling ordinary numbers, that is, the primitive numbers of our language. We will tag these with the symbol scheme-number. e arithmetic operations in this package are the primitive arithmetic procedures (so there is no need to deﬁne extra procedures to handle the untagged numbers). Since these operations each take two arguments, they are installed in the table keyed by the list (scheme-number scheme-number): (define (install-scheme-number-package) (define (tag x) (attach-tag 'scheme-number x)) (put 'add '(scheme-number scheme-number) (lambda (x y) (tag (+ x y)))) (put 'sub '(scheme-number scheme-number) (lambda (x y) (tag (- x y)))) (put 'mul '(scheme-number scheme-number) (lambda (x y) (tag (* x y)))) (put 'div '(scheme-number scheme-number) (lambda (x y) (tag (/ x y)))) (put 'make 'scheme-number (lambda (x) (tag x))) 'done) Users of the Scheme-number package will create (tagged) ordinary num- bers by means of the procedure: (define (make-scheme-number n) ((get 'make 'scheme-number) n)) Now that the framework of the generic arithmetic system is in place, we can readily include new kinds of numbers. Here is a package that performs rational arithmetic. Notice that, as a beneﬁt of additivity, we 256 can use without modiﬁcation the rational-number code from Section 2.1.1 as the internal procedures in the package: (define (install-rational-package) ;; internal procedures (define (numer x) (car x)) (define (denom x) (cdr x)) (define (make-rat n d) (let ((g (gcd n d))) (cons (/ n g) (/ d g)))) (define (add-rat x y) (make-rat (+ (* (numer x) (denom y)) (* (numer y) (denom x))) (* (denom x) (denom y)))) (define (sub-rat x y) (make-rat (- (* (numer x) (denom y)) (* (numer y) (denom x))) (* (denom x) (denom y)))) (define (mul-rat x y) (make-rat (* (numer x) (numer y)) (* (denom x) (denom y)))) (define (div-rat x y) (make-rat (* (numer x) (denom y)) (* (denom x) (numer y)))) ;; interface to rest of the system (define (tag x) (attach-tag 'rational x)) (put 'add '(rational rational) (lambda (x y) (tag (add-rat x y)))) (put 'sub '(rational rational) (lambda (x y) (tag (sub-rat x y)))) (put 'mul '(rational rational) (lambda (x y) (tag (mul-rat x y)))) (put 'div '(rational rational) (lambda (x y) (tag (div-rat x y)))) 257 (put 'make 'rational (lambda (n d) (tag (make-rat n d)))) 'done) (define (make-rational n d) ((get 'make 'rational) n d)) We can install a similar package to handle complex numbers, using the tag complex. In creating the package, we extract from the table the op- erations make-from-real-imag and make-from-mag-ang that were de- ﬁned by the rectangular and polar packages. Additivity permits us to use, as the internal operations, the same add-complex, sub-complex, mul-complex, and div-complex procedures from Section 2.4.1. (define (install-complex-package) ;; imported procedures from rectangular and polar packages (define (make-from-real-imag x y) ((get 'make-from-real-imag 'rectangular) x y)) (define (make-from-mag-ang r a) ((get 'make-from-mag-ang 'polar) r a)) ;; internal procedures (define (add-complex z1 z2) (make-from-real-imag (+ (real-part z1) (real-part z2)) (+ (imag-part z1) (imag-part z2)))) (define (sub-complex z1 z2) (make-from-real-imag (- (real-part z1) (real-part z2)) (- (imag-part z1) (imag-part z2)))) (define (mul-complex z1 z2) (make-from-mag-ang (* (magnitude z1) (magnitude z2)) (+ (angle z1) (angle z2)))) (define (div-complex z1 z2) (make-from-mag-ang (/ (magnitude z1) (magnitude z2)) (- (angle z1) (angle z2)))) ;; interface to rest of the system (define (tag z) (attach-tag 'complex z)) 258 (put 'add '(complex complex) (lambda (z1 z2) (tag (add-complex z1 z2)))) (put 'sub '(complex complex) (lambda (z1 z2) (tag (sub-complex z1 z2)))) (put 'mul '(complex complex) (lambda (z1 z2) (tag (mul-complex z1 z2)))) (put 'div '(complex complex) (lambda (z1 z2) (tag (div-complex z1 z2)))) (put 'make-from-real-imag 'complex (lambda (x y) (tag (make-from-real-imag x y)))) (put 'make-from-mag-ang 'complex (lambda (r a) (tag (make-from-mag-ang r a)))) 'done) Programs outside the complex-number package can construct complex numbers either from real and imaginary parts or from magnitudes and angles. Notice how the underlying procedures, originally deﬁned in the rectangular and polar packages, are exported to the complex package, and exported from there to the outside world. (define (make-complex-from-real-imag x y) ((get 'make-from-real-imag 'complex) x y)) (define (make-complex-from-mag-ang r a) ((get 'make-from-mag-ang 'complex) r a)) What we have here is a two-level tag system. A typical complex num- ber, such as 3 + 4i in rectangular form, would be represented as shown in Figure 2.24. e outer tag (complex) is used to direct the number to the complex package. Once within the complex package, the next tag (rectangular) is used to direct the number to the rectangular package. In a large and complicated system there might be many levels, each in- terfaced with the next by means of generic operations. As a data object is passed “downward,” the outer tag that is used to direct it to the ap- 259 complex rectangular 3 4 Figure 2.24: Representation of 3 + 4i in rectangular form. propriate package is stripped oﬀ (by applying contents) and the next level of tag (if any) becomes visible to be used for further dispatching. In the above packages, we used add-rat, add-complex, and the other arithmetic procedures exactly as originally wrien. Once these deﬁni- tions are internal to diﬀerent installation procedures, however, they no longer need names that are distinct from each other: we could simply name them add, sub, mul, and div in both packages. Exercise 2.77: Louis Reasoner tries to evaluate the expres- sion (magnitude z) where z is the object shown in Figure 2.24. To his surprise, instead of the answer 5 he gets an error message from apply-generic, saying there is no method for the operation magnitude on the types (complex). He shows this interaction to Alyssa P. Hacker, who says “e problem is that the complex-number selectors were never deﬁned for complex numbers, just for polar and rectangular numbers. All you have to do to make this work is add the following to the complex package:” (put 'real-part '(complex) real-part) (put 'imag-part '(complex) imag-part) (put 'magnitude '(complex) magnitude) (put 'angle '(complex) angle) 260 Describe in detail why this works. As an example, trace through all the procedures called in evaluating the expres- sion (magnitude z) where z is the object shown in Figure 2.24. In particular, how many times is apply-generic in- voked? What procedure is dispatched to in each case? Exercise 2.78: e internal procedures in the scheme-number package are essentially nothing more than calls to the prim- itive procedures +, -, etc. It was not possible to use the prim- itives of the language directly because our type-tag system requires that each data object have a type aached to it. In fact, however, all Lisp implementations do have a type sys- tem, which they use internally. Primitive predicates such as symbol? and number? determine whether data objects have particular types. Modify the deﬁnitions of type-tag, contents, and attach-tag from Section 2.4.2 so that our generic system takes advantage of Scheme’s internal type system. at is to say, the system should work as before ex- cept that ordinary numbers should be represented simply as Scheme numbers rather than as pairs whose car is the symbol scheme-number. Exercise 2.79: Deﬁne a generic equality predicate equ? that tests the equality of two numbers, and install it in the generic arithmetic package. is operation should work for ordi- nary numbers, rational numbers, and complex numbers. Exercise 2.80: Deﬁne a generic predicate =zero? that tests if its argument is zero, and install it in the generic arith- metic package. is operation should work for ordinary numbers, rational numbers, and complex numbers. 261 2.5.2 Combining Data of Diﬀerent Types We have seen how to deﬁne a uniﬁed arithmetic system that encom- passes ordinary numbers, complex numbers, rational numbers, and any other type of number we might decide to invent, but we have ignored an important issue. e operations we have deﬁned so far treat the diﬀer- ent data types as being completely independent. us, there are separate packages for adding, say, two ordinary numbers, or two complex num- bers. What we have not yet considered is the fact that it is meaningful to deﬁne operations that cross the type boundaries, such as the addition of a complex number to an ordinary number. We have gone to great pains to introduce barriers between parts of our programs so that they can be developed and understood separately. We would like to introduce the cross-type operations in some carefully controlled way, so that we can support them without seriously violating our module boundaries. One way to handle cross-type operations is to design a diﬀerent pro- cedure for each possible combination of types for which the operation is valid. For example, we could extend the complex-number package so that it provides a procedure for adding complex numbers to ordinary numbers and installs this in the table using the tag (complex scheme- number):49 ;; to be included in the complex package (define (add-complex-to-schemenum z x) (make-from-real-imag (+ (real-part z) x) (imag-part z))) (put 'add '(complex scheme-number) (lambda (z x) (tag (add-complex-to-schemenum z x)))) is technique works, but it is cumbersome. With such a system, the cost of introducing a new type is not just the construction of the pack- 49 We also have to supply an almost identical procedure to handle the types (scheme- number complex). 262 age of procedures for that type but also the construction and installa- tion of the procedures that implement the cross-type operations. is can easily be much more code than is needed to deﬁne the operations on the type itself. e method also undermines our ability to combine separate packages additively, or at least to limit the extent to which the implementors of the individual packages need to take account of other packages. For instance, in the example above, it seems reasonable that handling mixed operations on complex numbers and ordinary numbers should be the responsibility of the complex-number package. Combin- ing rational numbers and complex numbers, however, might be done by the complex package, by the rational package, or by some third package that uses operations extracted from these two packages. Formulating coherent policies on the division of responsibility among packages can be an overwhelming task in designing systems with many packages and many cross-type operations. Coercion In the general situation of completely unrelated operations acting on completely unrelated types, implementing explicit cross-type operations, cumbersome though it may be, is the best that one can hope for. For- tunately, we can usually do beer by taking advantage of additional structure that may be latent in our type system. Oen the diﬀerent data types are not completely independent, and there may be ways by which objects of one type may be viewed as being of another type. is process is called coercion. For example, if we are asked to arithmetically combine an ordinary number with a complex number, we can view the ordinary number as a complex number whose imaginary part is zero. is trans- forms the problem to that of combining two complex numbers, which can be handled in the ordinary way by the complex-arithmetic package. 263 In general, we can implement this idea by designing coercion pro- cedures that transform an object of one type into an equivalent object of another type. Here is a typical coercion procedure, which transforms a given ordinary number to a complex number with that real part and zero imaginary part: (define (scheme-number->complex n) (make-complex-from-real-imag (contents n) 0)) We install these coercion procedures in a special coercion table, indexed under the names of the two types: (put-coercion 'scheme-number 'complex scheme-number->complex) (We assume that there are put-coercion and get-coercion procedures available for manipulating this table.) Generally some of the slots in the table will be empty, because it is not generally possible to coerce an ar- bitrary data object of each type into all other types. For example, there is no way to coerce an arbitrary complex number to an ordinary num- ber, so there will be no general complex->scheme-number procedure included in the table. Once the coercion table has been set up, we can handle coercion in a uniform manner by modifying the apply-generic procedure of Section 2.4.3. When asked to apply an operation, we ﬁrst check whether the operation is deﬁned for the arguments’ types, just as before. If so, we dispatch to the procedure found in the operation-and-type table. Otherwise, we try coercion. For simplicity, we consider only the case where there are two arguments.50 We check the coercion table to see if objects of the ﬁrst type can be coerced to the second type. If so, we 50 See Exercise 2.82 for generalizations. 264 coerce the ﬁrst argument and try the operation again. If objects of the ﬁrst type cannot in general be coerced to the second type, we try the coercion the other way around to see if there is a way to coerce the second argument to the type of the ﬁrst argument. Finally, if there is no known way to coerce either type to the other type, we give up. Here is the procedure: (define (apply-generic op . args) (let ((type-tags (map type-tag args))) (let ((proc (get op type-tags))) (if proc (apply proc (map contents args)) (if (= (length args) 2) (let ((type1 (car type-tags)) (type2 (cadr type-tags)) (a1 (car args)) (a2 (cadr args))) (let ((t1->t2 (get-coercion type1 type2)) (t2->t1 (get-coercion type2 type1))) (cond (t1->t2 (apply-generic op (t1->t2 a1) a2)) (t2->t1 (apply-generic op a1 (t2->t1 a2))) (else (error "No method for these types" (list op type-tags)))))) (error "No method for these types" (list op type-tags))))))) is coercion scheme has many advantages over the method of deﬁning explicit cross-type operations, as outlined above. Although we still need to write coercion procedures to relate the types (possibly n 2 procedures for a system with n types), we need to write only one procedure for each pair of types rather than a diﬀerent procedure for each collection 265 of types and each generic operation.51 What we are counting on here is the fact that the appropriate transformation between types depends only on the types themselves, not on the operation to be applied. On the other hand, there may be applications for which our coer- cion scheme is not general enough. Even when neither of the objects to be combined can be converted to the type of the other it may still be possible to perform the operation by converting both objects to a third type. In order to deal with such complexity and still preserve modular- ity in our programs, it is usually necessary to build systems that take advantage of still further structure in the relations among types, as we discuss next. Hierarchies of types e coercion scheme presented above relied on the existence of natural relations between pairs of types. Oen there is more “global” structure in how the diﬀerent types relate to each other. For instance, suppose we are building a generic arithmetic system to handle integers, rational numbers, real numbers, and complex numbers. In such a system, it is quite natural to regard an integer as a special kind of rational number, which is in turn a special kind of real number, which is in turn a special kind of complex number. What we actually have is a so-called hierarchy of types, in which, for example, integers are a subtype of rational num- 51 If we are clever, we can usually get by with fewer than n 2 coercion procedures. For instance, if we know how to convert from type 1 to type 2 and from type 2 to type 3, then we can use this knowledge to convert from type 1 to type 3. is can greatly decrease the number of coercion procedures we need to supply explicitly when we add a new type to the system. If we are willing to build the required amount of sophistication into our system, we can have it search the “graph” of relations among types and automatically generate those coercion procedures that can be inferred from the ones that are supplied explicitly. 266 complex real rational integer Figure 2.25: A tower of types. bers (i.e., any operation that can be applied to a rational number can automatically be applied to an integer). Conversely, we say that ratio- nal numbers form a supertype of integers. e particular hierarchy we have here is of a very simple kind, in which each type has at most one supertype and at most one subtype. Such a structure, called a tower, is illustrated in Figure 2.25. If we have a tower structure, then we can greatly simplify the prob- lem of adding a new type to the hierarchy, for we need only specify how the new type is embedded in the next supertype above it and how it is the supertype of the type below it. For example, if we want to add an integer to a complex number, we need not explicitly deﬁne a special coercion procedure integer->complex. Instead, we deﬁne how an inte- ger can be transformed into a rational number, how a rational number is transformed into a real number, and how a real number is transformed into a complex number. We then allow the system to transform the in- teger into a complex number through these steps and then add the two complex numbers. We can redesign our apply-generic procedure in the following way: For each type, we need to supply a raise procedure, which “raises” 267 objects of that type one level in the tower. en when the system is re- quired to operate on objects of diﬀerent types it can successively raise the lower types until all the objects are at the same level in the tower. (Exercise 2.83 and Exercise 2.84 concern the details of implementing such a strategy.) Another advantage of a tower is that we can easily implement the notion that every type “inherits” all operations deﬁned on a supertype. For instance, if we do not supply a special procedure for ﬁnding the real part of an integer, we should nevertheless expect that real-part will be deﬁned for integers by virtue of the fact that integers are a subtype of complex numbers. In a tower, we can arrange for this to happen in a uniform way by modifying apply-generic. If the required operation is not directly deﬁned for the type of the object given, we raise the object to its supertype and try again. We thus crawl up the tower, transforming our argument as we go, until we either ﬁnd a level at which the desired operation can be performed or hit the top (in which case we give up). Yet another advantage of a tower over a more general hierarchy is that it gives us a simple way to “lower” a data object to the simplest representation. For example, if we add 2 + 3i to 4 − 3i, it would be nice to obtain the answer as the integer 6 rather than as the complex num- ber 6 + 0i. Exercise 2.85 discusses a way to implement such a lowering operation. (e trick is that we need a general way to distinguish those objects that can be lowered, such as 6 + 0i, from those that cannot, such as 6 + 2i.) Inadequacies of hierarchies If the data types in our system can be naturally arranged in a tower, this greatly simpliﬁes the problems of dealing with generic operations on diﬀerent types, as we have seen. Unfortunately, this is usually not the case. Figure 2.26 illustrates a more complex arrangement of mixed types, 268 polygon quadrilateral triangle trapezoid isosceles right parallelogram kite triangle triangle rectangle rhombus equilateral isosceles triangle right triangle square Figure 2.26: Relations among types of geometric ﬁgures. this one showing relations among diﬀerent types of geometric ﬁgures. We see that, in general, a type may have more than one subtype. Tri- angles and quadrilaterals, for instance, are both subtypes of polygons. In addition, a type may have more than one supertype. For example, an isosceles right triangle may be regarded either as an isosceles trian- gle or as a right triangle. is multiple-supertypes issue is particularly thorny, since it means that there is no unique way to “raise” a type in the hierarchy. Finding the “correct” supertype in which to apply an opera- tion to an object may involve considerable searching through the entire type network on the part of a procedure such as apply-generic. Since there generally are multiple subtypes for a type, there is a similar prob- lem in coercing a value “down” the type hierarchy. Dealing with large numbers of interrelated types while still preserving modularity in the 269 design of large systems is very diﬃcult, and is an area of much current research.52 Exercise 2.81: Louis Reasoner has noticed that apply-generic may try to coerce the arguments to each other’s type even if they already have the same type. erefore, he reasons, we need to put procedures in the coercion table to coerce arguments of each type to their own type. For example, in addition to the scheme-number->complex coercion shown above, he would do: (define (scheme-number->scheme-number n) n) (define (complex->complex z) z) (put-coercion 'scheme-number 'scheme-number scheme-number->scheme-number) (put-coercion 'complex 'complex complex->complex) 52 is statement, which also appears in the ﬁrst edition of this book, is just as true now as it was when we wrote it twelve years ago. Developing a useful, general frame- work for expressing the relations among diﬀerent types of entities (what philosophers call “ontology”) seems intractably diﬃcult. e main diﬀerence between the confu- sion that existed ten years ago and the confusion that exists now is that now a va- riety of inadequate ontological theories have been embodied in a plethora of corre- spondingly inadequate programming languages. For example, much of the complexity of object-oriented programming languages—and the subtle and confusing diﬀerences among contemporary object-oriented languages—centers on the treatment of generic operations on interrelated types. Our own discussion of computational objects in Chap- ter 3 avoids these issues entirely. Readers familiar with object-oriented programming will notice that we have much to say in chapter 3 about local state, but we do not even mention “classes” or “inheritance.” In fact, we suspect that these problems cannot be ad- equately addressed in terms of computer-language design alone, without also drawing on work in knowledge representation and automated reasoning. 270 a. With Louis’s coercion procedures installed, what hap- pens if apply-generic is called with two arguments of type scheme-number or two arguments of type complex for an operation that is not found in the table for those types? For example, assume that we’ve deﬁned a generic exponentiation operation: (define (exp x y) (apply-generic 'exp x y)) and have put a procedure for exponentiation in the Scheme-number package but not in any other pack- age: ;; following added to Scheme-number package (put 'exp '(scheme-number scheme-number) (lambda (x y) (tag (expt x y)))) ; using primitive expt What happens if we call exp with two complex num- bers as arguments? b. Is Louis correct that something had to be done about coercion with arguments of the same type, or does apply-generic work correctly as is? c. Modify apply-generic so that it doesn’t try coercion if the two arguments have the same type. Exercise 2.82: Show how to generalize apply-generic to handle coercion in the general case of multiple arguments. One strategy is to aempt to coerce all the arguments to the type of the ﬁrst argument, then to the type of the sec- ond argument, and so on. Give an example of a situation 271 where this strategy (and likewise the two-argument ver- sion given above) is not suﬃciently general. (Hint: Con- sider the case where there are some suitable mixed-type operations present in the table that will not be tried.) Exercise 2.83: Suppose you are designing a generic arith- metic system for dealing with the tower of types shown in Figure 2.25: integer, rational, real, complex. For each type (except complex), design a procedure that raises objects of that type one level in the tower. Show how to install a generic raise operation that will work for each type (ex- cept complex). Exercise 2.84: Using the raise operation of Exercise 2.83, modify the apply-generic procedure so that it coerces its arguments to have the same type by the method of succes- sive raising, as discussed in this section. You will need to devise a way to test which of two types is higher in the tower. Do this in a manner that is “compatible” with the rest of the system and will not lead to problems in adding new levels to the tower. Exercise 2.85: is section mentioned a method for “sim- plifying” a data object by lowering it in the tower of types as far as possible. Design a procedure drop that accom- plishes this for the tower described in Exercise 2.83. e key is to decide, in some general way, whether an object can be lowered. For example, the complex number 1.5 + 0i can be lowered as far as real, the complex number 1 + 0i can be lowered as far as integer, and the complex number 272 2 + 3i cannot be lowered at all. Here is a plan for determin- ing whether an object can be lowered: Begin by deﬁning a generic operation project that “pushes” an object down in the tower. For example, projecting a complex number would involve throwing away the imaginary part. en a number can be dropped if, when we project it and raise the result back to the type we started with, we end up with something equal to what we started with. Show how to im- plement this idea in detail, by writing a drop procedure that drops an object as far as possible. You will need to design the various projection operations53 and install project as a generic operation in the system. You will also need to make use of a generic equality predicate, such as described in Exercise 2.79. Finally, use drop to rewrite apply-generic from Exercise 2.84 so that it “simpliﬁes” its answers. Exercise 2.86: Suppose we want to handle complex num- bers whose real parts, imaginary parts, magnitudes, and an- gles can be either ordinary numbers, rational numbers, or other numbers we might wish to add to the system. De- scribe and implement the changes to the system needed to accommodate this. You will have to deﬁne operations such as sine and cosine that are generic over ordinary numbers and rational numbers. 53 A real number can be projected to an integer using the round primitive, which returns the closest integer to its argument. 273 2.5.3 Example: Symbolic Algebra e manipulation of symbolic algebraic expressions is a complex pro- cess that illustrates many of the hardest problems that occur in the de- sign of large-scale systems. An algebraic expression, in general, can be viewed as a hierarchical structure, a tree of operators applied to operands. We can construct algebraic expressions by starting with a set of primitive objects, such as constants and variables, and combining these by means of algebraic operators, such as addition and multipli- cation. As in other languages, we form abstractions that enable us to refer to compound objects in simple terms. Typical abstractions in sym- bolic algebra are ideas such as linear combination, polynomial, rational function, or trigonometric function. We can regard these as compound “types,” which are oen useful for directing the processing of expres- sions. For example, we could describe the expression x 2 sin(y 2 + 1) + x cos 2y + cos(y 3 − 2y 2 ) as a polynomial in x with coeﬃcients that are trigonometric functions of polynomials in y whose coeﬃcients are integers. We will not aempt to develop a complete algebraic-manipulation system here. Such systems are exceedingly complex programs, embody- ing deep algebraic knowledge and elegant algorithms. What we will do is look at a simple but important part of algebraic manipulation: the arithmetic of polynomials. We will illustrate the kinds of decisions the designer of such a system faces, and how to apply the ideas of abstract data and generic operations to help organize this eﬀort. Arithmetic on polynomials Our ﬁrst task in designing a system for performing arithmetic on poly- nomials is to decide just what a polynomial is. Polynomials are normally 274 deﬁned relative to certain variables (the indeterminates of the polyno- mial). For simplicity, we will restrict ourselves to polynomials having just one indeterminate (univariate polynomials).54 We will deﬁne a poly- nomial to be a sum of terms, each of which is either a coeﬃcient, a power of the indeterminate, or a product of a coeﬃcient and a power of the indeterminate. A coeﬃcient is deﬁned as an algebraic expression that is not dependent upon the indeterminate of the polynomial. For example, 5x 2 + 3x + 7 is a simple polynomial in x, and (y 2 + 1)x 3 + (2y)x + 1 is a polynomial in x whose coeﬃcients are polynomials in y. Already we are skirting some thorny issues. Is the ﬁrst of these poly- nomials the same as the polynomial 5y 2 + 3y + 7, or not? A reasonable answer might be “yes, if we are considering a polynomial purely as a mathematical function, but no, if we are considering a polynomial to be a syntactic form.” e second polynomial is algebraically equivalent to a polynomial in y whose coeﬃcients are polynomials in x . Should our system recognize this, or not? Furthermore, there are other ways to represent a polynomial—for example, as a product of factors, or (for a univariate polynomial) as the set of roots, or as a listing of the values of the polynomial at a speciﬁed set of points.55 We can ﬁnesse these ques- 54 On the other hand, we will allow polynomials whose coeﬃcients are themselves polynomials in other variables. is will give us essentially the same representational power as a full multivariate system, although it does lead to coercion problems, as discussed below. 55 For univariate polynomials, giving the value of a polynomial at a given set of points can be a particularly good representation. is makes polynomial arithmetic extremely 275 tions by deciding that in our algebraic-manipulation system a “polyno- mial” will be a particular syntactic form, not its underlying mathemat- ical meaning. Now we must consider how to go about doing arithmetic on polyno- mials. In this simple system, we will consider only addition and multi- plication. Moreover, we will insist that two polynomials to be combined must have the same indeterminate. We will approach the design of our system by following the familiar discipline of data abstraction. We will represent polynomials using a data structure called a poly, which consists of a variable and a collection of terms. We assume that we have selectors variable and term-list that extract those parts from a poly and a constructor make-poly that assembles a poly from a given variable and a term list. A variable will be just a symbol, so we can use the same-variable? procedure of Section 2.3.2 to compare variables. e following procedures deﬁne addition and multiplication of polys: (define (add-poly p1 p2) (if (same-variable? (variable p1) (variable p2)) (make-poly (variable p1) (add-terms (term-list p1) (term-list p2))) (error "Polys not in same var: ADD-POLY" (list p1 p2)))) (define (mul-poly p1 p2) (if (same-variable? (variable p1) (variable p2)) (make-poly (variable p1) (mul-terms (term-list p1) (term-list p2))) (error "Polys not in same var: MUL-POLY" (list p1 p2)))) simple. To obtain, for example, the sum of two polynomials represented in this way, we need only add the values of the polynomials at corresponding points. To transform back to a more familiar representation, we can use the Lagrange interpolation formula, which shows how to recover the coeﬃcients of a polynomial of degree n given the values of the polynomial at n + 1 points. 276 To incorporate polynomials into our generic arithmetic system, we need to supply them with type tags. We’ll use the tag polynomial, and install appropriate operations on tagged polynomials in the operation table. We’ll embed all our code in an installation procedure for the polynomial package, similar to the ones in Section 2.5.1: (define (install-polynomial-package) ;; internal procedures ;; representation of poly (define (make-poly variable term-list) (cons variable term-list)) (define (variable p) (car p)) (define (term-list p) (cdr p)) ⟨procedures same-variable? and variable? from section 2.3.2⟩ ;; representation of terms and term lists ⟨procedures adjoin-term . . . coeff from text below ⟩ (define (add-poly p1 p2) . . .) ⟨procedures used by add-poly⟩ (define (mul-poly p1 p2) . . .) ⟨procedures used by mul-poly⟩ ;; interface to rest of the system (define (tag p) (attach-tag 'polynomial p)) (put 'add '(polynomial polynomial) (lambda (p1 p2) (tag (add-poly p1 p2)))) (put 'mul '(polynomial polynomial) (lambda (p1 p2) (tag (mul-poly p1 p2)))) (put 'make 'polynomial (lambda (var terms) (tag (make-poly var terms)))) 'done) Polynomial addition is performed termwise. Terms of the same order (i.e., with the same power of the indeterminate) must be combined. is is done by forming a new term of the same order whose coeﬃcient is the sum of the coeﬃcients of the addends. Terms in one addend for which 277 there are no terms of the same order in the other addend are simply accumulated into the sum polynomial being constructed. In order to manipulate term lists, we will assume that we have a constructor the-empty-termlist that returns an empty term list and a constructor adjoin-term that adjoins a new term to a term list. We will also assume that we have a predicate empty-termlist? that tells if a given term list is empty, a selector first-term that extracts the highest- order term from a term list, and a selector rest-terms that returns all but the highest-order term. To manipulate terms, we will suppose that we have a constructor make-term that constructs a term with given or- der and coeﬃcient, and selectors order and coeff that return, respec- tively, the order and the coeﬃcient of the term. ese operations allow us to consider both terms and term lists as data abstractions, whose concrete representations we can worry about separately. Here is the procedure that constructs the term list for the sum of two polynomials:56 (define (add-terms L1 L2) (cond ((empty-termlist? L1) L2) ((empty-termlist? L2) L1) (else (let ((t1 (first-term L1)) (t2 (first-term L2))) (cond ((> (order t1) (order t2)) (adjoin-term t1 (add-terms (rest-terms L1) L2))) ((< (order t1) (order t2)) 56 is operation is very much like the ordered union-set operation we developed in Exercise 2.62. In fact, if we think of the terms of the polynomial as a set ordered according to the power of the indeterminate, then the program that produces the term list for a sum is almost identical to union-set. 278 (adjoin-term t2 (add-terms L1 (rest-terms L2)))) (else (adjoin-term (make-term (order t1) (add (coeff t1) (coeff t2))) (add-terms (rest-terms L1) (rest-terms L2))))))))) e most important point to note here is that we used the generic ad- dition procedure add to add together the coeﬃcients of the terms being combined. is has powerful consequences, as we will see below. In order to multiply two term lists, we multiply each term of the ﬁrst list by all the terms of the other list, repeatedly using mul-term- by-all-terms, which multiplies a given term by all terms in a given term list. e resulting term lists (one for each term of the ﬁrst list) are accumulated into a sum. Multiplying two terms forms a term whose order is the sum of the orders of the factors and whose coeﬃcient is the product of the coeﬃcients of the factors: (define (mul-terms L1 L2) (if (empty-termlist? L1) (the-empty-termlist) (add-terms (mul-term-by-all-terms (first-term L1) L2) (mul-terms (rest-terms L1) L2)))) (define (mul-term-by-all-terms t1 L) (if (empty-termlist? L) (the-empty-termlist) (let ((t2 (first-term L))) (adjoin-term (make-term (+ (order t1) (order t2)) (mul (coeff t1) (coeff t2))) (mul-term-by-all-terms t1 (rest-terms L)))))) 279 is is really all there is to polynomial addition and multiplication. No- tice that, since we operate on terms using the generic procedures add and mul, our polynomial package is automatically able to handle any type of coeﬃcient that is known about by the generic arithmetic pack- age. If we include a coercion mechanism such as one of those discussed in Section 2.5.2, then we also are automatically able to handle operations on polynomials of diﬀerent coeﬃcient types, such as [ ] 2 4 2 2 [3x + (2 + 3i)x + 7] · x + x + (5 + 3i) . 3 Because we installed the polynomial addition and multiplication proce- dures add-poly and mul-poly in the generic arithmetic system as the add and mul operations for type polynomial, our system is also auto- matically able to handle polynomial operations such as [ ] [ ] (y + 1)x 2 + (y 2 + 1)x + (y − 1) · (y − 2)x + (y 3 + 7) . e reason is that when the system tries to combine coeﬃcients, it will dispatch through add and mul. Since the coeﬃcients are themselves polynomials (in y), these will be combined using add-poly and mul- poly. e result is a kind of “data-directed recursion” in which, for ex- ample, a call to mul-poly will result in recursive calls to mul-poly in order to multiply the coeﬃcients. If the coeﬃcients of the coeﬃcients were themselves polynomials (as might be used to represent polynomi- als in three variables), the data direction would ensure that the system would follow through another level of recursive calls, and so on through as many levels as the structure of the data dictates.57 57 Tomake this work completely smoothly, we should also add to our generic arith- metic system the ability to coerce a “number” to a polynomial by regarding it as a 280 Representing term lists Finally, we must confront the job of implementing a good representa- tion for term lists. A term list is, in eﬀect, a set of coeﬃcients keyed by the order of the term. Hence, any of the methods for representing sets, as discussed in Section 2.3.3, can be applied to this task. On the other hand, our procedures add-terms and mul-terms always access term lists sequentially from highest to lowest order. us, we will use some kind of ordered list representation. How should we structure the list that represents a term list? One consideration is the “density” of the polynomials we intend to manip- ulate. A polynomial is said to be dense if it has nonzero coeﬃcients in terms of most orders. If it has many zero terms it is said to be sparse. For example, A : x 5 + 2x 4 + 3x 2 − 2x − 5 is a dense polynomial, whereas B: x 100 + 2x 2 + 1 is sparse. e term lists of dense polynomials are most eﬃciently represented as lists of the coeﬃcients. For example, A above would be nicely rep- resented as (1 2 0 3 -2 -5). e order of a term in this representa- tion is the length of the sublist beginning with that term’s coeﬃcient, polynomial of degree zero whose coeﬃcient is the number. is is necessary if we are going to perform operations such as [x 2 + (y + 1)x + 5] + [x 2 + 2x + 1], which requires adding the coeﬃcient y + 1 to the coeﬃcient 2. 281 decremented by 1.58 is would be a terrible representation for a sparse polynomial such as B: ere would be a giant list of zeros punctuated by a few lonely nonzero terms. A more reasonable representation of the term list of a sparse polynomial is as a list of the nonzero terms, where each term is a list containing the order of the term and the coeﬃcient for that order. In such a scheme, polynomial B is eﬃciently represented as ((100 1) (2 2) (0 1)). As most polynomial manipulations are performed on sparse polynomials, we will use this method. We will as- sume that term lists are represented as lists of terms, arranged from highest-order to lowest-order term. Once we have made this decision, implementing the selectors and constructors for terms and term lists is straightforward:59 (define (adjoin-term term term-list) (if (=zero? (coeff term)) term-list (cons term term-list))) (define (the-empty-termlist) '()) (define (first-term term-list) (car term-list)) (define (rest-terms term-list) (cdr term-list)) (define (empty-termlist? term-list) (null? term-list)) (define (make-term order coeff) (list order coeff)) 58 In these polynomial examples, we assume that we have implemented the generic arithmetic system using the type mechanism suggested in Exercise 2.78. us, coeﬃ- cients that are ordinary numbers will be represented as the numbers themselves rather than as pairs whose car is the symbol scheme-number. 59 Although we are assuming that term lists are ordered, we have implemented ad- join-term to simply cons the new term onto the existing term list. We can get away with this so long as we guarantee that the procedures (such as add-terms) that use ad- join-term always call it with a higher-order term than appears in the list. If we did not want to make such a guarantee, we could have implemented adjoin-term to be simi- lar to the adjoin-set constructor for the ordered-list representation of sets (Exercise 2.61). 282 (define (order term) (car term)) (define (coeff term) (cadr term)) where =zero? is as deﬁned in Exercise 2.80. (See also Exercise 2.87 be- low.) Users of the polynomial package will create (tagged) polynomials by means of the procedure: (define (make-polynomial var terms) ((get 'make 'polynomial) var terms)) Exercise 2.87: Install =zero? for polynomials in the generic arithmetic package. is will allow adjoin-term to work for polynomials with coeﬃcients that are themselves poly- nomials. Exercise 2.88: Extend the polynomial system to include subtraction of polynomials. (Hint: You may ﬁnd it helpful to deﬁne a generic negation operation.) Exercise 2.89: Deﬁne procedures that implement the term- list representation described above as appropriate for dense polynomials. Exercise 2.90: Suppose we want to have a polynomial sys- tem that is eﬃcient for both sparse and dense polynomials. One way to do this is to allow both kinds of term-list repre- sentations in our system. e situation is analogous to the complex-number example of Section 2.4, where we allowed both rectangular and polar representations. To do this we must distinguish diﬀerent types of term lists and make the operations on term lists generic. Redesign the polynomial 283 system to implement this generalization. is is a major ef- fort, not a local change. Exercise 2.91: A univariate polynomial can be divided by another one to produce a polynomial quotient and a poly- nomial remainder. For example, x5 − 1 = x 3 + x , remainder x − 1. x2 − 1 Division can be performed via long division. at is, divide the highest-order term of the dividend by the highest-order term of the divisor. e result is the ﬁrst term of the quo- tient. Next, multiply the result by the divisor, subtract that from the dividend, and produce the rest of the answer by re- cursively dividing the diﬀerence by the divisor. Stop when the order of the divisor exceeds the order of the dividend and declare the dividend to be the remainder. Also, if the dividend ever becomes zero, return zero as both quotient and remainder. We can design a div-poly procedure on the model of add- poly and mul-poly. e procedure checks to see if the two polys have the same variable. If so, div-poly strips oﬀ the variable and passes the problem to div-terms, which per- forms the division operation on term lists. div-poly ﬁnally reaaches the variable to the result supplied by div-terms. It is convenient to design div-terms to compute both the quotient and the remainder of a division. div-terms can take two term lists as arguments and return a list of the quotient term list and the remainder term list. 284 Complete the following deﬁnition of div-terms by ﬁlling in the missing expressions. Use this to implement div-poly, which takes two polys as arguments and returns a list of the quotient and remainder polys. (define (div-terms L1 L2) (if (empty-termlist? L1) (list (the-empty-termlist) (the-empty-termlist)) (let ((t1 (first-term L1)) (t2 (first-term L2))) (if (> (order t2) (order t1)) (list (the-empty-termlist) L1) (let ((new-c (div (coeff t1) (coeff t2))) (new-o (- (order t1) (order t2)))) (let ((rest-of-result ⟨compute rest of result recursively ⟩ )) ⟨form complete result⟩ )))))) Hierarchies of types in symbolic algebra Our polynomial system illustrates how objects of one type (polynomi- als) may in fact be complex objects that have objects of many diﬀerent types as parts. is poses no real diﬃculty in deﬁning generic opera- tions. We need only install appropriate generic operations for perform- ing the necessary manipulations of the parts of the compound types. In fact, we saw that polynomials form a kind of “recursive data abstrac- tion,” in that parts of a polynomial may themselves be polynomials. Our generic operations and our data-directed programming style can handle this complication without much trouble. On the other hand, polynomial algebra is a system for which the data types cannot be naturally arranged in a tower. For instance, it is possible to have polynomials in x whose coeﬃcients are polynomials in y. It is also possible to have polynomials in y whose coeﬃcients are 285 polynomials in x. Neither of these types is “above” the other in any natural way, yet it is oen necessary to add together elements from each set. ere are several ways to do this. One possibility is to convert one polynomial to the type of the other by expanding and rearrang- ing terms so that both polynomials have the same principal variable. One can impose a towerlike structure on this by ordering the variables and thus always converting any polynomial to a “canonical form” with the highest-priority variable dominant and the lower-priority variables buried in the coeﬃcients. is strategy works fairly well, except that the conversion may expand a polynomial unnecessarily, making it hard to read and perhaps less eﬃcient to work with. e tower strategy is certainly not natural for this domain or for any domain where the user can invent new types dynamically using old types in various combining forms, such as trigonometric functions, power series, and integrals. It should not be surprising that controlling coercion is a serious problem in the design of large-scale algebraic-manipulation systems. Much of the complexity of such systems is concerned with relationships among diverse types. Indeed, it is fair to say that we do not yet com- pletely understand coercion. In fact, we do not yet completely under- stand the concept of a data type. Nevertheless, what we know provides us with powerful structuring and modularity principles to support the design of large systems. Exercise 2.92: By imposing an ordering on variables, ex- tend the polynomial package so that addition and multipli- cation of polynomials works for polynomials in diﬀerent variables. (is is not easy!) 286 Extended exercise: Rational functions We can extend our generic arithmetic system to include rational func- tions. ese are “fractions” whose numerator and denominator are poly- nomials, such as x +1 . x3 − 1 e system should be able to add, subtract, multiply, and divide rational functions, and to perform such computations as x +1 x x 3 + 2x 2 + 3x + 1 + = . x3 − 1 x2 − 1 x4 + x3 − x − 1 (Here the sum has been simpliﬁed by removing common factors. Ordi- nary “cross multiplication” would have produced a fourth-degree poly- nomial over a ﬁh-degree polynomial.) If we modify our rational-arithmetic package so that it uses generic operations, then it will do what we want, except for the problem of reducing fractions to lowest terms. Exercise 2.93: Modify the rational-arithmetic package to use generic operations, but change make-rat so that it does not aempt to reduce fractions to lowest terms. Test your system by calling make-rational on two polynomials to produce a rational function: (define p1 (make-polynomial 'x '((2 1) (0 1)))) (define p2 (make-polynomial 'x '((3 1) (0 1)))) (define rf (make-rational p2 p1)) Now add rf to itself, using add. You will observe that this addition procedure does not reduce fractions to lowest terms. 287 We can reduce polynomial fractions to lowest terms using the same idea we used with integers: modifying make-rat to divide both the numera- tor and the denominator by their greatest common divisor. e notion of “greatest common divisor” makes sense for polynomials. In fact, we can compute the of two polynomials using essentially the same Euclid’s Algorithm that works for integers.60 e integer version is (define (gcd a b) (if (= b 0) a (gcd b (remainder a b)))) Using this, we could make the obvious modiﬁcation to deﬁne a operation that works on term lists: (define (gcd-terms a b) (if (empty-termlist? b) a (gcd-terms b (remainder-terms a b)))) where remainder-terms picks out the remainder component of the list returned by the term-list division operation div-terms that was imple- mented in Exercise 2.91. 60 e fact that Euclid’s Algorithm works for polynomials is formalized in algebra by saying that polynomials form a kind of algebraic domain called a Euclidean ring. A Euclidean ring is a domain that admits addition, subtraction, and commutative mul- tiplication, together with a way of assigning to each element x of the ring a positive integer “measure” m(x ) with the properties that m(xy) ≥ m(x ) for any nonzero x and y and that, given any x and y, there exists a q such that y = qx + r and either r = 0 or m(r ) < m(x ). From an abstract point of view, this is what is needed to prove that Euclid’s Algorithm works. For the domain of integers, the measure m of an integer is the absolute value of the integer itself. For the domain of polynomials, the measure of a polynomial is its degree. 288 Exercise 2.94: Using div-terms, implement the procedure remainder-terms and use this to deﬁne gcd-terms as above. Now write a procedure gcd-poly that computes the poly- nomial of two polys. (e procedure should signal an error if the two polys are not in the same variable.) Install in the system a generic operation greatest-common-divisor that reduces to gcd-poly for polynomials and to ordinary gcd for ordinary numbers. As a test, try (define p1 (make-polynomial 'x '((4 1) (3 -1) (2 -2) (1 2)))) (define p2 (make-polynomial 'x '((3 1) (1 -1)))) (greatest-common-divisor p1 p2) and check your result by hand. Exercise 2.95: Deﬁne P1 , P2 , and P3 to be the polynomials P1 : x 2 − 2x + 1, P2 : 11x 2 + 7, P3 : 13x + 5. Now deﬁne Q 1 to be the product of P1 and P2 and Q 2 to be the product of P1 and P3 , and use greatest-common- divisor (Exercise 2.94) to compute the of Q 1 and Q 2 . Note that the answer is not the same as P1 . is example in- troduces noninteger operations into the computation, caus- ing diﬃculties with the algorithm.61 To understand 61 In an implementation like Scheme, this produces a polynomial that is indeed a divisor of Q 1 and Q 2 , but with rational coeﬃcients. In many other Scheme systems, in which division of integers can produce limited-precision decimal numbers, we may fail to get a valid divisor. 289 what is happening, try tracing gcd-terms while comput- ing the or try performing the division by hand. We can solve the problem exhibited in Exercise 2.95 if we use the follow- ing modiﬁcation of the algorithm (which really works only in the case of polynomials with integer coeﬃcients). Before performing any polynomial division in the computation, we multiply the dividend by an integer constant factor, chosen to guarantee that no fractions will arise during the division process. Our answer will thus diﬀer from the actual by an integer constant factor, but this does not maer in the case of reducing rational functions to lowest terms; the will be used to divide both the numerator and denominator, so the integer constant factor will cancel out. More precisely, if P and Q are polynomials, let O 1 be the order of P (i.e., the order of the largest term of P ) and let O 2 be the order of Q. Let c be the leading coeﬃcient of Q. en it can be shown that, if we multiply P by the integerizing factor c 1+O 1 −O 2 , the resulting polynomial can be divided by Q by using the div-terms algorithm without introducing any fractions. e operation of multiplying the dividend by this constant and then dividing is sometimes called the pseudodivision of P by Q. e remainder of the division is called the pseudoremainder. Exercise 2.96: a. Implement the procedure pseudoremainder-terms, which is just like remainder-terms except that it multiplies the dividend by the integerizing factor described above before calling div-terms. Modify gcd-terms to use pseudoremainder-terms, and verify that greatest- common-divisor now produces an answer with inte- ger coeﬃcients on the example in Exercise 2.95. 290 b. e now has integer coeﬃcients, but they are larger than those of P1 . Modify gcd-terms so that it removes common factors from the coeﬃcients of the answer by dividing all the coeﬃcients by their (inte- ger) greatest common divisor. us, here is how to reduce a rational function to lowest terms: • Compute the of the numerator and denominator, using the version of gcd-terms from Exercise 2.96. • When you obtain the , multiply both numerator and denomi- nator by the same integerizing factor before dividing through by the , so that division by the will not introduce any nonin- teger coeﬃcients. As the factor you can use the leading coeﬃcient of the raised to the power 1 + O 1 − O 2 , where O 2 is the order of the and O 1 is the maximum of the orders of the numerator and denominator. is will ensure that dividing the numerator and denominator by the will not introduce any fractions. • e result of this operation will be a numerator and denominator with integer coeﬃcients. e coeﬃcients will normally be very large because of all of the integerizing factors, so the last step is to remove the redundant factors by computing the (integer) greatest common divisor of all the coeﬃcients of the numerator and the denominator and dividing through by this factor. Exercise 2.97: a. Implement this algorithm as a procedure reduce-terms that takes two term lists n and d as arguments and re- 291 turns a list nn, dd, which are n and d reduced to low- est terms via the algorithm given above. Also write a procedure reduce-poly, analogous to add-poly, that checks to see if the two polys have the same variable. If so, reduce-poly strips oﬀ the variable and passes the problem to reduce-terms, then reaaches the vari- able to the two term lists supplied by reduce-terms. b. Deﬁne a procedure analogous to reduce-terms that does what the original make-rat did for integers: (define (reduce-integers n d) (let ((g (gcd n d))) (list (/ n g) (/ d g)))) and deﬁne reduce as a generic operation that calls apply-generic to dispatch to either reduce-poly (for polynomial arguments) or reduce-integers (for scheme- number arguments). You can now easily make the rational- arithmetic package reduce fractions to lowest terms by having make-rat call reduce before combining the given numerator and denominator to form a ratio- nal number. e system now handles rational expres- sions in either integers or polynomials. To test your program, try the example at the beginning of this ex- tended exercise: (define p1 (make-polynomial 'x '((1 1) (0 1)))) (define p2 (make-polynomial 'x '((3 1) (0 -1)))) (define p3 (make-polynomial 'x '((1 1)))) (define p4 (make-polynomial 'x '((2 1) (0 -1)))) (define rf1 (make-rational p1 p2)) (define rf2 (make-rational p3 p4)) 292 (add rf1 rf2) See if you get the correct answer, correctly reduced to lowest terms. e computation is at the heart of any system that does opera- tions on rational functions. e algorithm used above, although mathe- matically straightforward, is extremely slow. e slowness is due partly to the large number of division operations and partly to the enormous size of the intermediate coeﬃcients generated by the pseudodivisions. One of the active areas in the development of algebraic-manipulation systems is the design of beer algorithms for computing polynomial s.62 62 One extremely eﬃcient and elegant method for computing polynomial s was discovered by Richard Zippel (1979). e method is a probabilistic algorithm, as is the fast test for primality that we discussed in Chapter 1. Zippel’s book (Zippel 1993) de- scribes this method, together with other ways to compute polynomial s. 293 Modularity, Objects, and State Mεταβάλλον ὰναπαύεται (Even while it changes, it stands still.) —Heraclitus Plus ça change, plus c’est la même chose. —Alphonse Karr T introduced the basic elements from which programs are made. We saw how primitive procedures and primi- tive data are combined to construct compound entities, and we learned that abstraction is vital in helping us to cope with the complexity of large systems. But these tools are not suﬃcient for designing programs. Eﬀective program synthesis also requires organizational principles that can guide us in formulating the overall design of a program. In partic- ular, we need strategies to help us structure large systems so that they 294 will be modular, that is, so that they can be divided “naturally” into co- herent parts that can be separately developed and maintained. One powerful design strategy, which is particularly appropriate to the construction of programs for modeling physical systems, is to base the structure of our programs on the structure of the system being mod- eled. For each object in the system, we construct a corresponding com- putational object. For each system action, we deﬁne a symbolic opera- tion in our computational model. Our hope in using this strategy is that extending the model to accommodate new objects or new actions will require no strategic changes to the program, only the addition of the new symbolic analogs of those objects or actions. If we have been suc- cessful in our system organization, then to add a new feature or debug an old one we will have to work on only a localized part of the system. To a large extent, then, the way we organize a large program is dic- tated by our perception of the system to be modeled. In this chapter we will investigate two prominent organizational strategies arising from two rather diﬀerent “world views” of the structure of systems. e ﬁrst organizational strategy concentrates on objects, viewing a large system as a collection of distinct objects whose behaviors may change over time. An alternative organizational strategy concentrates on the streams of information that ﬂow in the system, much as an electrical engineer views a signal-processing system. Both the object-based approach and the stream-processing approach raise signiﬁcant linguistic issues in programming. With objects, we must be concerned with how a computational object can change and yet main- tain its identity. is will force us to abandon our old substitution model of computation (Section 1.1.5) in favor of a more mechanistic but less theoretically tractable environment model of computation. e diﬃcul- ties of dealing with objects, change, and identity are a fundamental con- 295 sequence of the need to grapple with time in our computational models. ese diﬃculties become even greater when we allow the possibility of concurrent execution of programs. e stream approach can be most fully exploited when we decouple simulated time in our model from the order of the events that take place in the computer during evaluation. We will accomplish this using a technique known as delayed evaluation. 3.1 Assignment and Local State We ordinarily view the world as populated by independent objects, each of which has a state that changes over time. An object is said to “have state” if its behavior is inﬂuenced by its history. A bank account, for example, has state in that the answer to the question “Can I withdraw $100?” depends upon the history of deposit and withdrawal transac- tions. We can characterize an object’s state by one or more state vari- ables, which among them maintain enough information about history to determine the object’s current behavior. In a simple banking system, we could characterize the state of an account by a current balance rather than by remembering the entire history of account transactions. In a system composed of many objects, the objects are rarely com- pletely independent. Each may inﬂuence the states of others through interactions, which serve to couple the state variables of one object to those of other objects. Indeed, the view that a system is composed of separate objects is most useful when the state variables of the system can be grouped into closely coupled subsystems that are only loosely coupled to other subsystems. is view of a system can be a powerful framework for organizing computational models of the system. For such a model to be modular, it should be decomposed into computational objects that model the actual 296 objects in the system. Each computational object must have its own lo- cal state variables describing the actual object’s state. Since the states of objects in the system being modeled change over time, the state vari- ables of the corresponding computational objects must also change. If we choose to model the ﬂow of time in the system by the elapsed time in the computer, then we must have a way to construct computational objects whose behaviors change as our programs run. In particular, if we wish to model state variables by ordinary symbolic names in the programming language, then the language must provide an assignment operator to enable us to change the value associated with a name. 3.1.1 Local State Variables To illustrate what we mean by having a computational object with time- varying state, let us model the situation of withdrawing money from a bank account. We will do this using a procedure withdraw, which takes as argument an amount to be withdrawn. If there is enough money in the account to accommodate the withdrawal, then withdraw should re- turn the balance remaining aer the withdrawal. Otherwise, withdraw should return the message Insuﬃcient funds. For example, if we begin with $100 in the account, we should obtain the following sequence of responses using withdraw: (withdraw 25) 75 (withdraw 25) 50 (withdraw 60) "Insufficient funds" (withdraw 15) 35 297 Observe that the expression (withdraw 25), evaluated twice, yields diﬀerent values. is is a new kind of behavior for a procedure. Until now, all our procedures could be viewed as speciﬁcations for comput- ing mathematical functions. A call to a procedure computed the value of the function applied to the given arguments, and two calls to the same procedure with the same arguments always produced the same result.1 To implement withdraw, we can use a variable balance to indicate the balance of money in the account and deﬁne withdraw as a procedure that accesses balance. e withdraw procedure checks to see if balance is at least as large as the requested amount. If so, withdraw decrements balance by amount and returns the new value of balance. Otherwise, withdraw returns the Insuﬃcient funds message. Here are the deﬁnitions of balance and withdraw: (define balance 100) (define (withdraw amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")) Decrementing balance is accomplished by the expression (set! balance (- balance amount)) is uses the set! special form, whose syntax is (set! ⟨name⟩ ⟨new-value⟩) 1 Actually, this is not quite true. One exception was the random-number generator in Section 1.2.6. Another exception involved the operation/type tables we introduced in Section 2.4.3, where the values of two calls to get with the same arguments depended on intervening calls to put. On the other hand, until we introduce assignment, we have no way to create such procedures ourselves. 298 Here ⟨name ⟩ is a symbol and ⟨new-value ⟩ is any expression. set! changes ⟨name ⟩ so that its value is the result obtained by evaluating ⟨new-value ⟩. In the case at hand, we are changing balance so that its new value will be the result of subtracting amount from the previous value of balance.2 withdraw also uses the begin special form to cause two expressions to be evaluated in the case where the if test is true: ﬁrst decrementing balance and then returning the value of balance. In general, evaluating the expression (begin ⟨exp1 ⟩ ⟨exp2 ⟩ . . . ⟨expk ⟩) causes the expressions ⟨ exp1 ⟩ through ⟨ expk ⟩ to be evaluated in se- quence and the value of the ﬁnal expression ⟨ expk ⟩ to be returned as the value of the entire begin form.3 Although withdraw works as desired, the variable balance presents a problem. As speciﬁed above, balance is a name deﬁned in the global environment and is freely accessible to be examined or modiﬁed by any procedure. It would be much beer if we could somehow make balance internal to withdraw, so that withdraw would be the only procedure that could access balance directly and any other procedure could access balance only indirectly (through calls to withdraw). is would more accurately model the notion that balance is a local state variable used 2 e value of a set! expression is implementation-dependent. set! should be used only for its eﬀect, not for its value. e name set! reﬂects a naming convention used in Scheme: Operations that change the values of variables (or that change data structures, as we will see in Section 3.3) are given names that end with an exclamation point. is is similar to the convention of designating predicates by names that end with a question mark. 3 We have already used begin implicitly in our programs, because in Scheme the body of a procedure can be a sequence of expressions. Also, the ⟨consequent ⟩ part of each clause in a cond expression can be a sequence of expressions rather than a single expression. 299 by withdraw to keep track of the state of the account. We can make balance internal to withdraw by rewriting the deﬁ- nition as follows: (define new-withdraw (let ((balance 100)) (lambda (amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")))) What we have done here is use let to establish an environment with a local variable balance, bound to the initial value 100. Within this local environment, we use lambda to create a procedure that takes amount as an argument and behaves like our previous withdraw procedure. is procedure—returned as the result of evaluating the let expression—is new-withdraw, which behaves in precisely the same way as withdraw but whose variable balance is not accessible by any other procedure.4 Combining set! with local variables is the general programming technique we will use for constructing computational objects with lo- cal state. Unfortunately, using this technique raises a serious problem: When we ﬁrst introduced procedures, we also introduced the substi- tution model of evaluation (Section 1.1.5) to provide an interpretation of what procedure application means. We said that applying a proce- dure should be interpreted as evaluating the body of the procedure with the formal parameters replaced by their values. e trouble is that, as 4 In programming-language jargon, the variable balance is said to be encapsulated within the new-withdraw procedure. Encapsulation reﬂects the general system-design principle known as the hiding principle: One can make a system more modular and ro- bust by protecting parts of the system from each other; that is, by providing information access only to those parts of the system that have a “need to know.” 300 soon as we introduce assignment into our language, substitution is no longer an adequate model of procedure application. (We will see why this is so in Section 3.1.3.) As a consequence, we technically have at this point no way to understand why the new-withdraw procedure be- haves as claimed above. In order to really understand a procedure such as new-withdraw, we will need to develop a new model of procedure ap- plication. In Section 3.2 we will introduce such a model, together with an explanation of set! and local variables. First, however, we examine some variations on the theme established by new-withdraw. e following procedure, make-withdraw, creates “withdrawal pro- cessors.” e formal parameter balance in make-withdraw speciﬁes the initial amount of money in the account.5 (define (make-withdraw balance) (lambda (amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds"))) make-withdraw can be used as follows to create two objects W1 and W2: (define W1 (make-withdraw 100)) (define W2 (make-withdraw 100)) (W1 50) 50 (W2 70) 30 5 In contrast with new-withdraw above, we do not have to use let to make balance a local variable, since formal parameters are already local. is will be clearer aer the discussion of the environment model of evaluation in Section 3.2. (See also Exercise 3.10.) 301 (W2 40) "Insufficient funds" (W1 40) 10 Observe that W1 and W2 are completely independent objects, each with its own local state variable balance. Withdrawals from one do not aﬀect the other. We can also create objects that handle deposits as well as with- drawals, and thus we can represent simple bank accounts. Here is a procedure that returns a “bank-account object” with a speciﬁed initial balance: (define (make-account balance) (define (withdraw amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")) (define (deposit amount) (set! balance (+ balance amount)) balance) (define (dispatch m) (cond ((eq? m 'withdraw) withdraw) ((eq? m 'deposit) deposit) (else (error "Unknown request: MAKE-ACCOUNT" m)))) dispatch) Each call to make-account sets up an environment with a local state variable balance. Within this environment, make-account deﬁnes pro- cedures deposit and withdraw that access balance and an additional procedure dispatch that takes a “message” as input and returns one of 302 the two local procedures. e dispatch procedure itself is returned as the value that represents the bank-account object. is is precisely the message-passing style of programming that we saw in Section 2.4.3, al- though here we are using it in conjunction with the ability to modify local variables. make-account can be used as follows: (define acc (make-account 100)) ((acc 'withdraw) 50) 50 ((acc 'withdraw) 60) "Insufficient funds" ((acc 'deposit) 40) 90 ((acc 'withdraw) 60) 30 Each call to acc returns the locally deﬁned deposit or withdraw pro- cedure, which is then applied to the speciﬁed amount. As was the case with make-withdraw, another call to make-account (define acc2 (make-account 100)) will produce a completely separate account object, which maintains its own local balance. Exercise 3.1: An accumulator is a procedure that is called repeatedly with a single numeric argument and accumu- lates its arguments into a sum. Each time it is called, it returns the currently accumulated sum. Write a procedure make-accumulator that generates accumulators, each main- taining an independent sum. e input to make-accumulator should specify the initial value of the sum; for example 303 (define A (make-accumulator 5)) (A 10) 15 (A 10) 25 Exercise 3.2: In soware-testing applications, it is useful to be able to count the number of times a given procedure is called during the course of a computation. Write a pro- cedure make-monitored that takes as input a procedure, f, that itself takes one input. e result returned by make- monitored is a third procedure, say mf, that keeps track of the number of times it has been called by maintaining an internal counter. If the input to mf is the special symbol how-many-calls?, then mf returns the value of the counter. If the input is the special symbol reset-count, then mf re- sets the counter to zero. For any other input, mf returns the result of calling f on that input and increments the counter. For instance, we could make a monitored version of the sqrt procedure: (define s (make-monitored sqrt)) (s 100) 10 (s 'how-many-calls?) 1 Exercise 3.3: Modify the make-account procedure so that it creates password-protected accounts. at is, make-account should take a symbol as an additional argument, as in (define acc (make-account 100 'secret-password)) 304 e resulting account object should process a request only if it is accompanied by the password with which the ac- count was created, and should otherwise return a complaint: ((acc 'secret-password 'withdraw) 40) 60 ((acc 'some-other-password 'deposit) 50) "Incorrect password" Exercise 3.4: Modify the make-account procedure of Ex- ercise 3.3 by adding another local state variable so that, if an account is accessed more than seven consecutive times with an incorrect password, it invokes the procedure call- the-cops. 3.1.2 The Benefits of Introducing Assignment As we shall see, introducing assignment into our programming lan- guage leads us into a thicket of diﬃcult conceptual issues. Nevertheless, viewing systems as collections of objects with local state is a powerful technique for maintaining a modular design. As a simple example, con- sider the design of a procedure rand that, whenever it is called, returns an integer chosen at random. It is not at all clear what is meant by “chosen at random.” What we presumably want is for successive calls to rand to produce a sequence of numbers that has statistical properties of uniform distribution. We will not discuss methods for generating suitable sequences here. Rather, let us assume that we have a procedure rand-update that has the property that if we start with a given number x 1 and form x 2 = (rand-update x 1 ) x 3 = (rand-update x 2 ) 305 then the sequence of values x 1 , x 2 , x 3 , . . . will have the desired statistical properties.6 We can implement rand as a procedure with a local state variable x that is initialized to some ﬁxed value random-init. Each call to rand computes rand-update of the current value of x, returns this as the random number, and also stores this as the new value of x. (define rand (let ((x random-init)) (lambda () (set! x (rand-update x)) x))) Of course, we could generate the same sequence of random numbers without using assignment by simply calling rand-update directly. How- ever, this would mean that any part of our program that used random numbers would have to explicitly remember the current value of x to be passed as an argument to rand-update. To realize what an annoy- ance this would be, consider using random numbers to implement a technique called Monte Carlo simulation. e Monte Carlo method consists of choosing sample experiments at random from a large set and then making deductions on the basis of 6 One common way to implement rand-update is to use the rule that x is updated to ax + b modulo m, where a, b, and m are appropriately chosen integers. Chapter 3 of Knuth 1981 includes an extensive discussion of techniques for generating sequences of random numbers and establishing their statistical properties. Notice that the rand- update procedure computes a mathematical function: Given the same input twice, it produces the same output. erefore, the number sequence produced by rand-update certainly is not “random,” if by “random” we insist that each number in the sequence is unrelated to the preceding number. e relation between “real randomness” and so- called pseudo-random sequences, which are produced by well-determined computations and yet have suitable statistical properties, is a complex question involving diﬃcult issues in mathematics and philosophy. Kolmogorov, Solomonoﬀ, and Chaitin have made great progress in clarifying these issues; a discussion can be found in Chaitin 1975. 306 the probabilities estimated from tabulating the results of those experi- ments. For example, we can approximate π using the fact that 6/π 2 is the probability that two integers chosen at random will have no fac- tors in common; that is, that their greatest common divisor will be 1.7 To obtain the approximation to π , we perform a large number of ex- periments. In each experiment we choose two integers at random and perform a test to see if their is 1. e fraction of times that the test is passed gives us our estimate of 6/π 2 , and from this we obtain our approximation to π . e heart of our program is a procedure monte-carlo, which takes as arguments the number of times to try an experiment, together with the experiment, represented as a no-argument procedure that will re- turn either true or false each time it is run. monte-carlo runs the exper- iment for the designated number of trials and returns a number telling the fraction of the trials in which the experiment was found to be true. (define (estimate-pi trials) (sqrt (/ 6 (monte-carlo trials cesaro-test)))) (define (cesaro-test) (= (gcd (rand) (rand)) 1)) (define (monte-carlo trials experiment) (define (iter trials-remaining trials-passed) (cond ((= trials-remaining 0) (/ trials-passed trials)) ((experiment) (iter (- trials-remaining 1) (+ trials-passed 1))) (else 7 is theorem is due to E. Cesàro. See section 4.5.2 of Knuth 1981 for a discussion and a proof. 307 (iter (- trials-remaining 1) trials-passed)))) (iter trials 0)) Now let us try the same computation using rand-update directly rather than rand, the way we would be forced to proceed if we did not use assignment to model local state: (define (estimate-pi trials) (sqrt (/ 6 (random-gcd-test trials random-init)))) (define (random-gcd-test trials initial-x) (define (iter trials-remaining trials-passed x) (let ((x1 (rand-update x))) (let ((x2 (rand-update x1))) (cond ((= trials-remaining 0) (/ trials-passed trials)) ((= (gcd x1 x2) 1) (iter (- trials-remaining 1) (+ trials-passed 1) x2)) (else (iter (- trials-remaining 1) trials-passed x2)))))) (iter trials 0 initial-x)) While the program is still simple, it betrays some painful breaches of modularity. In our ﬁrst version of the program, using rand, we can ex- press the Monte Carlo method directly as a general monte-carlo proce- dure that takes as an argument an arbitrary experiment procedure. In our second version of the program, with no local state for the random- number generator, random-gcd-test must explicitly manipulate the ran- dom numbers x1 and x2 and recycle x2 through the iterative loop as the new input to rand-update. is explicit handling of the random 308 numbers intertwines the structure of accumulating test results with the fact that our particular experiment uses two random numbers, whereas other Monte Carlo experiments might use one random number or three. Even the top-level procedure estimate-pi has to be concerned with supplying an initial random number. e fact that the random-number generator’s insides are leaking out into other parts of the program makes it diﬃcult for us to isolate the Monte Carlo idea so that it can be applied to other tasks. In the ﬁrst version of the program, assignment encapsu- lates the state of the random-number generator within the rand proce- dure, so that the details of random-number generation remain indepen- dent of the rest of the program. e general phenomenon illustrated by the Monte Carlo example is this: From the point of view of one part of a complex process, the other parts appear to change with time. ey have hidden time-varying local state. If we wish to write computer programs whose structure reﬂects this decomposition, we make computational objects (such as bank ac- counts and random-number generators) whose behavior changes with time. We model state with local state variables, and we model the changes of state with assignments to those variables. It is tempting to conclude this discussion by saying that, by intro- ducing assignment and the technique of hiding state in local variables, we are able to structure systems in a more modular fashion than if all state had to be manipulated explicitly, by passing additional parameters. Unfortunately, as we shall see, the story is not so simple. Exercise 3.5: Monte Carlo integration is a method of esti- mating deﬁnite integrals by means of Monte Carlo simula- tion. Consider computing the area of a region of space de- scribed by a predicate P (x , y) that is true for points (x , y) in the region and false for points not in the region. For 309 example, the region contained within a circle of radius 3 centered at (5, 7) is described by the predicate that tests whether (x − 5)2 + (y − 7)2 ≤ 32 . To estimate the area of the region described by such a predicate, begin by choosing a rectangle that contains the region. For example, a rectangle with diagonally opposite corners at (2, 4) and (8, 10) con- tains the circle above. e desired integral is the area of that portion of the rectangle that lies in the region. We can estimate the integral by picking, at random, points (x , y) that lie in the rectangle, and testing P (x , y) for each point to determine whether the point lies in the region. If we try this with many points, then the fraction of points that fall in the region should give an estimate of the proportion of the rectangle that lies in the region. Hence, multiplying this fraction by the area of the entire rectangle should produce an estimate of the integral. Implement Monte Carlo integration as a procedure estimate- integral that takes as arguments a predicate P, upper and lower bounds x1, x2, y1, and y2 for the rectangle, and the number of trials to perform in order to produce the esti- mate. Your procedure should use the same monte-carlo procedure that was used above to estimate π . Use your estimate- integral to produce an estimate of π by measuring the area of a unit circle. You will ﬁnd it useful to have a procedure that returns a number chosen at random from a given range. e follow- ing random-in-range procedure implements this in terms of the random procedure used in Section 1.2.6, which re- 310 turns a nonnegative number less than its input.8 (define (random-in-range low high) (let ((range (- high low))) (+ low (random range)))) Exercise 3.6: It is useful to be able to reset a random-number generator to produce a sequence starting from a given value. Design a new rand procedure that is called with an ar- gument that is either the symbol generate or the symbol reset and behaves as follows: (rand 'generate) produces a new random number; ((rand 'reset) ⟨new-value ⟩) re- sets the internal state variable to the designated ⟨new-value ⟩. us, by reseing the state, one can generate repeatable se- quences. ese are very handy to have when testing and debugging programs that use random numbers. 3.1.3 The Costs of Introducing Assignment As we have seen, the set! operation enables us to model objects that have local state. However, this advantage comes at a price. Our pro- gramming language can no longer be interpreted in terms of the sub- stitution model of procedure application that we introduced in Section 1.1.5. Moreover, no simple model with “nice” mathematical properties can be an adequate framework for dealing with objects and assignment in programming languages. So long as we do not use assignments, two evaluations of the same procedure with the same arguments will produce the same result, so 8 Scheme provides such a procedure. If random is given an exact integer (as in Section 1.2.6) it returns an exact integer, but if it is given a decimal value (as in this exercise) it returns a decimal value. 311 that procedures can be viewed as computing mathematical functions. Programming without any use of assignments, as we did throughout the ﬁrst two chapters of this book, is accordingly known as functional programming. To understand how assignment complicates maers, consider a sim- pliﬁed version of the make-withdraw procedure of Section 3.1.1 that does not bother to check for an insuﬃcient amount: (define (make-simplified-withdraw balance) (lambda (amount) (set! balance (- balance amount)) balance)) (define W (make-simplified-withdraw 25)) (W 20) 5 (W 10) -5 Compare this procedure with the following make-decrementer proce- dure, which does not use set!: (define (make-decrementer balance) (lambda (amount) (- balance amount))) make-decrementer returns a procedure that subtracts its input from a designated amount balance, but there is no accumulated eﬀect over successive calls, as with make-simplified-withdraw: (define D (make-decrementer 25)) (D 20) 5 (D 10) 15 312 We can use the substitution model to explain how make-decrementer works. For instance, let us analyze the evaluation of the expression ((make-decrementer 25) 20) We ﬁrst simplify the operator of the combination by substituting 25 for balance in the body of make-decrementer. is reduces the expression to ((lambda (amount) (- 25 amount)) 20) Now we apply the operator by substituting 20 for amount in the body of the lambda expression: (- 25 20) e ﬁnal answer is 5. Observe, however, what happens if we aempt a similar substitution analysis with make-simplified-withdraw: ((make-simplified-withdraw 25) 20) We ﬁrst simplify the operator by substituting 25 for balance in the body of make-simplified-withdraw. is reduces the expression to9 ((lambda (amount) (set! balance (- 25 amount)) 25) 20) Now we apply the operator by substituting 20 for amount in the body of the lambda expression: (set! balance (- 25 20)) 25 If we adhered to the substitution model, we would have to say that the meaning of the procedure application is to ﬁrst set balance to 5 and then 9 We don’t substitute for the occurrence of balance in the set! expression because the ⟨name ⟩ in a set! is not evaluated. If we did substitute for it, we would get (set! 25 (- 25 amount)), which makes no sense. 313 return 25 as the value of the expression. is gets the wrong answer. In order to get the correct answer, we would have to somehow distinguish the ﬁrst occurrence of balance (before the eﬀect of the set!) from the second occurrence of balance (aer the eﬀect of the set!), and the substitution model cannot do this. e trouble here is that substitution is based ultimately on the no- tion that the symbols in our language are essentially names for values. But as soon as we introduce set! and the idea that the value of a vari- able can change, a variable can no longer be simply a name. Now a variable somehow refers to a place where a value can be stored, and the value stored at this place can change. In Section 3.2 we will see how environments play this role of “place” in our computational model. Sameness and change e issue surfacing here is more profound than the mere breakdown of a particular model of computation. As soon as we introduce change into our computational models, many notions that were previously straight- forward become problematical. Consider the concept of two things be- ing “the same.” Suppose we call make-decrementer twice with the same argument to create two procedures: (define D1 (make-decrementer 25)) (define D2 (make-decrementer 25)) Are D1 and D2 the same? An acceptable answer is yes, because D1 and D2 have the same computational behavior—each is a procedure that sub- tracts its input from 25. In fact, D1 could be substituted for D2 in any computation without changing the result. Contrast this with making two calls to make-simplified-withdraw: 314 (define W1 (make-simplified-withdraw 25)) (define W2 (make-simplified-withdraw 25)) Are W1 and W2 the same? Surely not, because calls to W1 and W2 have distinct eﬀects, as shown by the following sequence of interactions: (W1 20) 5 (W1 20) -15 (W2 20) 5 Even though W1 and W2 are “equal” in the sense that they are both cre- ated by evaluating the same expression, (make-simplified-withdraw 25), it is not true that W1 could be substituted for W2 in any expression without changing the result of evaluating the expression. A language that supports the concept that “equals can be substituted for equals” in an expression without changing the value of the expres- sion is said to be referentially transparent. Referential transparency is violated when we include set! in our computer language. is makes it tricky to determine when we can simplify expressions by substituting equivalent expressions. Consequently, reasoning about programs that use assignment becomes drastically more diﬃcult. Once we forgo referential transparency, the notion of what it means for computational objects to be “the same” becomes diﬃcult to capture in a formal way. Indeed, the meaning of “same” in the real world that our programs model is hardly clear in itself. In general, we can determine that two apparently identical objects are indeed “the same one” only by modifying one object and then observing whether the other object has changed in the same way. But how can we tell if an object has “changed” other than by observing the “same” object twice and seeing whether 315 some property of the object diﬀers from one observation to the next? us, we cannot determine “change” without some a priori notion of “sameness,” and we cannot determine sameness without observing the eﬀects of change. As an example of how this issue arises in programming, consider the situation where Peter and Paul have a bank account with $100 in it. ere is a substantial diﬀerence between modeling this as (define peter-acc (make-account 100)) (define paul-acc (make-account 100)) and modeling it as (define peter-acc (make-account 100)) (define paul-acc peter-acc) In the ﬁrst situation, the two bank accounts are distinct. Transactions made by Peter will not aﬀect Paul’s account, and vice versa. In the sec- ond situation, however, we have deﬁned paul-acc to be the same thing as peter-acc. In eﬀect, Peter and Paul now have a joint bank account, and if Peter makes a withdrawal from peter-acc Paul will observe less money in paul-acc. ese two similar but distinct situations can cause confusion in building computational models. With the shared account, in particular, it can be especially confusing that there is one object (the bank account) that has two diﬀerent names (peter-acc and paul-acc); if we are searching for all the places in our program where paul-acc can be changed, we must remember to look also at things that change peter-acc.10 10 e phenomenon of a single computational object being accessed by more than one name is known as aliasing. e joint bank account situation illustrates a very simple example of an alias. In Section 3.3 we will see much more complex examples, such as “distinct” compound data structures that share parts. Bugs can occur in our programs 316 With reference to the above remarks on “sameness” and “change,” observe that if Peter and Paul could only examine their bank balances, and could not perform operations that changed the balance, then the is- sue of whether the two accounts are distinct would be moot. In general, so long as we never modify data objects, we can regard a compound data object to be precisely the totality of its pieces. For example, a ratio- nal number is determined by giving its numerator and its denominator. But this view is no longer valid in the presence of change, where a com- pound data object has an “identity” that is something diﬀerent from the pieces of which it is composed. A bank account is still “the same” bank account even if we change the balance by making a withdrawal; con- versely, we could have two diﬀerent bank accounts with the same state information. is complication is a consequence, not of our program- ming language, but of our perception of a bank account as an object. We do not, for example, ordinarily regard a rational number as a change- able object with identity, such that we could change the numerator and still have “the same” rational number. Pitfalls of imperative programming In contrast to functional programming, programming that makes ex- tensive use of assignment is known as imperative programming. In ad- dition to raising complications about computational models, programs wrien in imperative style are susceptible to bugs that cannot occur in functional programs. For example, recall the iterative factorial program if we forget that a change to an object may also, as a “side eﬀect,” change a “diﬀerent” object because the two “diﬀerent” objects are actually a single object appearing under diﬀerent aliases. ese so-called side-eﬀect bugs are so diﬃcult to locate and to analyze that some people have proposed that programming languages be designed in such a way as to not allow side eﬀects or aliasing (Lampson et al. 1981; Morris et al. 1980). 317 from Section 1.2.1: (define (factorial n) (define (iter product counter) (if (> counter n) product (iter (* counter product) (+ counter 1)))) (iter 1 1)) Instead of passing arguments in the internal iterative loop, we could adopt a more imperative style by using explicit assignment to update the values of the variables product and counter: (define (factorial n) (let ((product 1) (counter 1)) (define (iter) (if (> counter n) product (begin (set! product (* counter product)) (set! counter (+ counter 1)) (iter)))) (iter))) is does not change the results produced by the program, but it does introduce a subtle trap. How do we decide the order of the assignments? As it happens, the program is correct as wrien. But writing the assign- ments in the opposite order (set! counter (+ counter 1)) (set! product (* counter product)) would have produced a diﬀerent, incorrect result. In general, program- ming with assignment forces us to carefully consider the relative orders of the assignments to make sure that each statement is using the correct 318 version of the variables that have been changed. is issue simply does not arise in functional programs.11 e complexity of imperative programs becomes even worse if we consider applications in which several processes execute concurrently. We will return to this in Section 3.4. First, however, we will address the issue of providing a computational model for expressions that involve assignment, and explore the uses of objects with local state in designing simulations. Exercise 3.7: Consider the bank account objects created by make-account, with the password modiﬁcation described in Exercise 3.3. Suppose that our banking system requires the ability to make joint accounts. Deﬁne a procedure make- joint that accomplishes this. make-joint should take three arguments. e ﬁrst is a password-protected account. e second argument must match the password with which the account was deﬁned in order for the make-joint operation to proceed. e third argument is a new password. make- joint is to create an additional access to the original ac- count using the new password. For example, if peter-acc is a bank account with password open-sesame, then (define paul-acc (make-joint peter-acc 'open-sesame 'rosebud)) 11 In view of this, it is ironic that introductory programming is most oen taught in a highly imperative style. is may be a vestige of a belief, common throughout the 1960s and 1970s, that programs that call procedures must inherently be less eﬃ- cient than programs that perform assignments. (Steele 1977 debunks this argument.) Alternatively it may reﬂect a view that step-by-step assignment is easier for beginners to visualize than procedure call. Whatever the reason, it oen saddles beginning pro- grammers with “should I set this variable before or aer that one” concerns that can complicate programming and obscure the important ideas. 319 will allow one to make transactions on peter-acc using the name paul-acc and the password rosebud. You may wish to modify your solution to Exercise 3.3 to accommodate this new feature. Exercise 3.8: When we deﬁned the evaluation model in Section 1.1.3, we said that the ﬁrst step in evaluating an expression is to evaluate its subexpressions. But we never speciﬁed the order in which the subexpressions should be evaluated (e.g., le to right or right to le). When we in- troduce assignment, the order in which the arguments to a procedure are evaluated can make a diﬀerence to the result. Deﬁne a simple procedure f such that evaluating (+ (f 0) (f 1)) will return 0 if the arguments to + are evaluated from le to right but will return 1 if the arguments are evaluated from right to le. 3.2 The Environment Model of Evaluation When we introduced compound procedures in Chapter 1, we used the substitution model of evaluation (Section 1.1.5) to deﬁne what is meant by applying a procedure to arguments: • To apply a compound procedure to arguments, evaluate the body of the procedure with each formal parameter replaced by the cor- responding argument. Once we admit assignment into our programming language, such a def- inition is no longer adequate. In particular, Section 3.1.3 argued that, in 320 the presence of assignment, a variable can no longer be considered to be merely a name for a value. Rather, a variable must somehow designate a “place” in which values can be stored. In our new model of evaluation, these places will be maintained in structures called environments. An environment is a sequence of frames. Each frame is a table (pos- sibly empty) of bindings, which associate variable names with their cor- responding values. (A single frame may contain at most one binding for any variable.) Each frame also has a pointer to its enclosing environ- ment, unless, for the purposes of discussion, the frame is considered to be global. e value of a variable with respect to an environment is the value given by the binding of the variable in the ﬁrst frame in the en- vironment that contains a binding for that variable. If no frame in the sequence speciﬁes a binding for the variable, then the variable is said to be unbound in the environment. Figure 3.1 shows a simple environment structure consisting of three frames, labeled I, II, and III. In the diagram, A, B, C, and D are pointers to environments. C and D point to the same environment. e variables z and x are bound in frame II, while y and x are bound in frame I. e value of x in environment D is 3. e value of x with respect to environment B is also 3. is is determined as follows: We examine the ﬁrst frame in the sequence (frame III) and do not ﬁnd a binding for x, so we proceed to the enclosing environment D and ﬁnd the binding in frame I. On the other hand, the value of x in environment A is 7, because the ﬁrst frame in the sequence (frame II) contains a binding of x to 7. With respect to environment A, the binding of x to 7 in frame II is said to shadow the binding of x to 3 in frame I. e environment is crucial to the evaluation process, because it de- termines the context in which an expression should be evaluated. In- deed, one could say that expressions in a programming language do 321 I x:3 y:5 C D II III z:6 m:1 x:7 y:2 A B Figure 3.1: A simple environment structure. not, in themselves, have any meaning. Rather, an expression acquires a meaning only with respect to some environment in which it is evalu- ated. Even the interpretation of an expression as straightforward as (+ 1 1) depends on an understanding that one is operating in a context in which + is the symbol for addition. us, in our model of evaluation we will always speak of evaluating an expression with respect to some envi- ronment. To describe interactions with the interpreter, we will suppose that there is a global environment, consisting of a single frame (with no enclosing environment) that includes values for the symbols associated with the primitive procedures. For example, the idea that + is the sym- bol for addition is captured by saying that the symbol + is bound in the global environment to the primitive addition procedure. 3.2.1 The Rules for Evaluation e overall speciﬁcation of how the interpreter evaluates a combination remains the same as when we ﬁrst introduced it in Section 1.1.3: 322 • To evaluate a combination: 1. Evaluate the subexpressions of the combination.12 2. Apply the value of the operator subexpression to the values of the operand subexpressions. e environment model of evaluation replaces the substitution model in specifying what it means to apply a compound procedure to arguments. In the environment model of evaluation, a procedure is always a pair consisting of some code and a pointer to an environment. Procedures are created in one way only: by evaluating a λ-expression. is produces a procedure whose code is obtained from the text of the λ-expression and whose environment is the environment in which the λ-expression was evaluated to produce the procedure. For example, consider the pro- cedure deﬁnition (define (square x) (* x x)) evaluated in the global environment. e procedure deﬁnition syntax is just syntactic sugar for an underlying implicit λ-expression. It would have been equivalent to have used (define square (lambda (x) (* x x))) 12 Assignment introduces a subtlety into step 1 of the evaluation rule. As shown in Exercise 3.8, the presence of assignment allows us to write expressions that will produce diﬀerent values depending on the order in which the subexpressions in a combination are evaluated. us, to be precise, we should specify an evaluation order in step 1 (e.g., le to right or right to le). However, this order should always be considered to be an implementation detail, and one should never write programs that depend on some particular order. For instance, a sophisticated compiler might optimize a program by varying the order in which subexpressions are evaluated. 323 global other variables env square: (define (square x) (* x x)) parameters: x body: (* x x) Figure 3.2: Environment structure produced by evaluating (define (square x) (* x x)) in the global environment. which evaluates (lambda (x) (* x x)) and binds square to the re- sulting value, all in the global environment. Figure 3.2 shows the result of evaluating this define expression. e procedure object is a pair whose code speciﬁes that the procedure has one formal parameter, namely x, and a procedure body (* x x). e environment part of the procedure is a pointer to the global envi- ronment, since that is the environment in which the λ-expression was evaluated to produce the procedure. A new binding, which associates the procedure object with the symbol square, has been added to the global frame. In general, define creates deﬁnitions by adding bindings to frames. Now that we have seen how procedures are created, we can describe how procedures are applied. e environment model speciﬁes: To ap- ply a procedure to arguments, create a new environment containing a frame that binds the parameters to the values of the arguments. e en- closing environment of this frame is the environment speciﬁed by the 324 global other variables env square: (square 5) E1 x:5 (* x x) parameters: x body: (* x x) Figure 3.3: Environment created by evaluating (square 5) in the global environment. procedure. Now, within this new environment, evaluate the procedure body. To show how this rule is followed, Figure 3.3 illustrates the environ- ment structure created by evaluating the expression (square 5) in the global environment, where square is the procedure generated in Figure 3.2. Applying the procedure results in the creation of a new environ- ment, labeled E1 in the ﬁgure, that begins with a frame in which x, the formal parameter for the procedure, is bound to the argument 5. e pointer leading upward from this frame shows that the frame’s enclos- ing environment is the global environment. e global environment is chosen here, because this is the environment that is indicated as part of the square procedure object. Within E1, we evaluate the body of the procedure, (* x x). Since the value of x in E1 is 5, the result is (* 5 5), or 25. e environment model of procedure application can be summa- rized by two rules: 325 • A procedure object is applied to a set of arguments by construct- ing a frame, binding the formal parameters of the procedure to the arguments of the call, and then evaluating the body of the proce- dure in the context of the new environment constructed. e new frame has as its enclosing environment the environment part of the procedure object being applied. • A procedure is created by evaluating a λ-expression relative to a given environment. e resulting procedure object is a pair con- sisting of the text of the λ-expression and a pointer to the envi- ronment in which the procedure was created. We also specify that deﬁning a symbol using define creates a binding in the current environment frame and assigns to the symbol the indi- cated value.13 Finally, we specify the behavior of set!, the operation that forced us to introduce the environment model in the ﬁrst place. Evaluating the expression (set! ⟨variable ⟩ ⟨value ⟩) in some environ- ment locates the binding of the variable in the environment and changes that binding to indicate the new value. at is, one ﬁnds the ﬁrst frame in the environment that contains a binding for the variable and modi- ﬁes that frame. If the variable is unbound in the environment, then set! signals an error. ese evaluation rules, though considerably more complex than the substitution model, are still reasonably straightforward. Moreover, the evaluation model, though abstract, provides a correct description of 13 If there is already a binding for the variable in the current frame, then the binding is changed. is is convenient because it allows redeﬁnition of symbols; however, it also means that define can be used to change values, and this brings up the issues of assign- ment without explicitly using set!. Because of this, some people prefer redeﬁnitions of existing symbols to signal errors or warnings. 326 how the interpreter evaluates expressions. In Chapter 4 we shall see how this model can serve as a blueprint for implementing a working interpreter. e following sections elaborate the details of the model by analyzing some illustrative programs. 3.2.2 Applying Simple Procedures When we introduced the substitution model in Section 1.1.5 we showed how the combination (f 5) evaluates to 136, given the following pro- cedure deﬁnitions: (define (square x) (* x x)) (define (sum-of-squares x y) (+ (square x) (square y))) (define (f a) (sum-of-squares (+ a 1) (* a 2))) We can analyze the same example using the environment model. Figure 3.4 shows the three procedure objects created by evaluating the deﬁni- tions of f, square, and sum-of-squares in the global environment. Each procedure object consists of some code, together with a pointer to the global environment. In Figure 3.5 we see the environment structure created by evaluat- ing the expression (f 5). e call to f creates a new environment E1 beginning with a frame in which a, the formal parameter of f, is bound to the argument 5. In E1, we evaluate the body of f: (sum-of-squares (+ a 1) (* a 2)) To evaluate this combination, we ﬁrst evaluate the subexpressions. e ﬁrst subexpression, sum-of-squares, has a value that is a procedure ob- ject. (Notice how this value is found: We ﬁrst look in the ﬁrst frame of 327 sum-of-squares: global square: env f: parameters: a parameters: x parameters: x, y body: (sum-of-squares body: (* x x) body: (+ (square x) (+ a 1) (square y)) (* a 2)) Figure 3.4: Procedure objects in the global frame. E1, which contains no binding for sum-of-squares. en we proceed to the enclosing environment, i.e. the global environment, and ﬁnd the binding shown in Figure 3.4.) e other two subexpressions are evalu- ated by applying the primitive operations + and * to evaluate the two combinations (+ a 1) and (* a 2) to obtain 6 and 10, respectively. Now we apply the procedure object sum-of-squares to the argu- ments 6 and 10. is results in a new environment E2 in which the formal parameters x and y are bound to the arguments. Within E2 we evaluate the combination (+ (square x) (square y)). is leads us to evaluate (square x), where square is found in the global frame and x is 6. Once again, we set up a new environment, E3, in which x is bound to 6, and within this we evaluate the body of square, which is (* x x). Also as part of applying sum-of-squares, we must evaluate the subex- pression (square y), where y is 10. is second call to square creates another environment, E4, in which x, the formal parameter of square, is bound to 10. And within E4 we must evaluate (* x x). 328 global env (f 5) a:5 x:6 x:6 x:10 E1 E2 E3 E4 y:10 (sum-of-squares (+ (square x) (* x x) (* x x) (+ a 1) (square y)) (* a 2)) Figure 3.5: Environments created by evaluating (f 5) us- ing the procedures in Figure 3.4. e important point to observe is that each call to square creates a new environment containing a binding for x. We can see here how the diﬀerent frames serve to keep separate the diﬀerent local variables all named x. Notice that each frame created by square points to the global environment, since this is the environment indicated by the square pro- cedure object. Aer the subexpressions are evaluated, the results are returned. e values generated by the two calls to square are added by sum-of-squares, and this result is returned by f. Since our focus here is on the environ- ment structures, we will not dwell on how these returned values are passed from call to call; however, this is also an important aspect of the evaluation process, and we will return to it in detail in Chapter 5. Exercise 3.9: In Section 1.2.1 we used the substitution model to analyze two procedures for computing factorials, a recur- sive version 329 (define (factorial n) (if (= n 1) 1 (* n (factorial (- n 1))))) and an iterative version (define (factorial n) (fact-iter 1 1 n)) (define (fact-iter product counter max-count) (if (> counter max-count) product (fact-iter (* counter product) (+ counter 1) max-count))) Show the environment structures created by evaluating (factorial 6) using each version of the factorial pro- cedure.14 3.2.3 Frames as the Repository of Local State We can turn to the environment model to see how procedures and as- signment can be used to represent objects with local state. As an exam- ple, consider the “withdrawal processor” from Section 3.1.1 created by calling the procedure (define (make-withdraw balance) (lambda (amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds"))) 14 e environment model will not clarify our claim in Section 1.2.1 that the inter- preter can execute a procedure such as fact-iter in a constant amount of space using tail recursion. We will discuss tail recursion when we deal with the control structure of the interpreter in Section 5.4. 330 global make-withdraw: env parameters: balance body: (lambda (amount) (if (>= balance amount) (begin (set! balance (-- balance amount)) balance) "insufficient funds")) Figure 3.6: Result of deﬁning make-withdraw in the global environment. Let us describe the evaluation of (define W1 (make-withdraw 100)) followed by (W1 50) 50 Figure 3.6 shows the result of deﬁning the make-withdraw procedure in the global environment. is produces a procedure object that contains a pointer to the global environment. So far, this is no diﬀerent from the examples we have already seen, except that the body of the procedure is itself a λ-expression. e interesting part of the computation happens when we apply the procedure make-withdraw to an argument: (define W1 (make-withdraw 100)) 331 global make-withdraw: env W1: E1 balance: 100 parameters: balance body: ... parameters: amount body: (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "insufficient funds") Figure 3.7: Result of evaluating (define W1 (make-withdraw 100)). We begin, as usual, by seing up an environment E1 in which the formal parameter balance is bound to the argument 100. Within this environ- ment, we evaluate the body of make-withdraw, namely the λ-expression. is constructs a new procedure object, whose code is as speciﬁed by the lambda and whose environment is E1, the environment in which the lambda was evaluated to produce the procedure. e resulting pro- cedure object is the value returned by the call to make-withdraw. is is bound to W1 in the global environment, since the define itself is be- ing evaluated in the global environment. Figure 3.7 shows the resulting environment structure. Now we can analyze what happens when W1 is applied to an argu- ment: (W1 50) 50 332 global make-withdraw: ... env W1: Here is the balance E1 balance: 100 that will be changed by the set! amount: 50 parameters: amount (if (>= balance amount) body: ... (begin (set! balance (- balance amount)) balance) "insufficient funds") Figure 3.8: Environments created by applying the procedure object W1. We begin by constructing a frame in which amount, the formal pa- rameter of W1, is bound to the argument 50. e crucial point to ob- serve is that this frame has as its enclosing environment not the global environment, but rather the environment E1, because this is the envi- ronment that is speciﬁed by the W1 procedure object. Within this new environment, we evaluate the body of the procedure: (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds") e resulting environment structure is shown in Figure 3.8. e expres- sion being evaluated references both amount and balance. amount will be found in the ﬁrst frame in the environment, while balance will be 333 global make-withdraw: ... env W1: E1 balance: 50 parameters: amount body: ... Figure 3.9: Environments aer the call to W1. found by following the enclosing-environment pointer to E1. When the set! is executed, the binding of balance in E1 is changed. At the completion of the call to W1, balance is 50, and the frame that contains balance is still pointed to by the procedure object W1. e frame that binds amount (in which we executed the code that changed balance) is no longer relevant, since the procedure call that constructed it has terminated, and there are no pointers to that frame from other parts of the environment. e next time W1 is called, this will build a new frame that binds amount and whose enclosing environment is E1. We see that E1 serves as the “place” that holds the local state variable for the procedure object W1. Figure 3.9 shows the situation aer the call to W1. Observe what happens when we create a second “withdraw” object by making another call to make-withdraw: (define W2 (make-withdraw 100)) 334 make-withdraw: ... global W2: env W1: E1 balance: 50 E2 balance: 100 parameters: amount body: ... Figure 3.10: Using (define W2 (make-withdraw 100)) to create a second object. is produces the environment structure of Figure 3.10, which shows that W2 is a procedure object, that is, a pair with some code and an en- vironment. e environment E2 for W2 was created by the call to make- withdraw. It contains a frame with its own local binding for balance. On the other hand, W1 and W2 have the same code: the code speciﬁed by the λ-expression in the body of make-withdraw.15 We see here why W1 and W2 behave as independent objects. Calls to W1 reference the state variable balance stored in E1, whereas calls to W2 reference the balance stored in E2. us, changes to the local state of one object do not aﬀect the other object. 15 Whether W1 and W2 share the same physical code stored in the computer, or whether they each keep a copy of the code, is a detail of the implementation. For the interpreter we implement in Chapter 4, the code is in fact shared. 335 Exercise 3.10: In the make-withdraw procedure, the local variable balance is created as a parameter of make-withdraw. We could also create the local state variable explicitly, us- ing let, as follows: (define (make-withdraw initial-amount) (let ((balance initial-amount)) (lambda (amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")))) Recall from Section 1.3.2 that let is simply syntactic sugar for a procedure call: (let ((⟨var⟩ ⟨exp⟩)) ⟨body⟩) is interpreted as an alternate syntax for ((lambda (⟨var⟩) ⟨body⟩) ⟨exp⟩) Use the environment model to analyze this alternate ver- sion of make-withdraw, drawing ﬁgures like the ones above to illustrate the interactions (define W1 (make-withdraw 100)) (W1 50) (define W2 (make-withdraw 100)) Show that the two versions of make-withdraw create ob- jects with the same behavior. How do the environment struc- tures diﬀer for the two versions? 336 3.2.4 Internal Definitions Section 1.1.8 introduced the idea that procedures can have internal def- initions, thus leading to a block structure as in the following procedure to compute square roots: (define (sqrt x) (define (good-enough? guess) (< (abs (- (square guess) x)) 0.001)) (define (improve guess) (average guess (/ x guess))) (define (sqrt-iter guess) (if (good-enough? guess) guess (sqrt-iter (improve guess)))) (sqrt-iter 1.0)) Now we can use the environment model to see why these internal deﬁ- nitions behave as desired. Figure 3.11 shows the point in the evaluation of the expression (sqrt 2) where the internal procedure good-enough? has been called for the ﬁrst time with guess equal to 1. Observe the structure of the environment. sqrt is a symbol in the global environment that is bound to a procedure object whose associ- ated environment is the global environment. When sqrt was called, a new environment E1 was formed, subordinate to the global environ- ment, in which the parameter x is bound to 2. e body of sqrt was then evaluated in E1. Since the ﬁrst expression in the body of sqrt is (define (good-enough? guess) (< (abs (- (square guess) x)) 0.001)) evaluating this expression deﬁned the procedure good-enough? in the environment E1. To be more precise, the symbol good-enough? was added to the ﬁrst frame of E1, bound to a procedure object whose asso- 337 global sqrt: env x:2 good-enough?: E1 improve: ... parameters: x sqrt-iter: ... body: (define good-enough? ...) (define improve ...) (define sqrt-iter ...) (sqrt-iter 1.0) E2 guess: 1 parameters: guess call to sqrt-iter body: (< (abs ...) ...) E3 guess: 1 call to good-enough? Figure 3.11: sqrt procedure with internal deﬁnitions. ciated environment is E1. Similarly, improve and sqrt-iter were de- ﬁned as procedures in E1. For conciseness, Figure 3.11 shows only the procedure object for good-enough?. Aer the local procedures were deﬁned, the expression (sqrt-iter 1.0) was evaluated, still in environment E1. So the procedure object bound to sqrt-iter in E1 was called with 1 as an argument. is cre- ated an environment E2 in which guess, the parameter of sqrt-iter, is bound to 1. sqrt-iter in turn called good-enough? with the value of guess (from E2) as the argument for good-enough?. is set up another 338 environment, E3, in which guess (the parameter of good-enough?) is bound to 1. Although sqrt-iter and good-enough? both have a pa- rameter named guess, these are two distinct local variables located in diﬀerent frames. Also, E2 and E3 both have E1 as their enclosing en- vironment, because the sqrt-iter and good-enough? procedures both have E1 as their environment part. One consequence of this is that the symbol x that appears in the body of good-enough? will reference the binding of x that appears in E1, namely the value of x with which the original sqrt procedure was called. e environment model thus explains the two key properties that make local procedure deﬁnitions a useful technique for modularizing programs: • e names of the local procedures do not interfere with names external to the enclosing procedure, because the local procedure names will be bound in the frame that the procedure creates when it is run, rather than being bound in the global environment. • e local procedures can access the arguments of the enclosing procedure, simply by using parameter names as free variables. is is because the body of the local procedure is evaluated in an environment that is subordinate to the evaluation environment for the enclosing procedure. Exercise 3.11: In Section 3.2.3 we saw how the environ- ment model described the behavior of procedures with local state. Now we have seen how internal deﬁnitions work. A typical message-passing procedure contains both of these aspects. Consider the bank account procedure of Section 3.1.1: 339 (define (make-account balance) (define (withdraw amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")) (define (deposit amount) (set! balance (+ balance amount)) balance) (define (dispatch m) (cond ((eq? m 'withdraw) withdraw) ((eq? m 'deposit) deposit) (else (error "Unknown request: MAKE-ACCOUNT" m)))) dispatch) Show the environment structure generated by the sequence of interactions (define acc (make-account 50)) ((acc 'deposit) 40) 90 ((acc 'withdraw) 60) 30 Where is the local state for acc kept? Suppose we deﬁne another account (define acc2 (make-account 100)) How are the local states for the two accounts kept distinct? Which parts of the environment structure are shared be- tween acc and acc2? 340 3.3 Modeling with Mutable Data Chapter 2 dealt with compound data as a means for constructing com- putational objects that have several parts, in order to model real-world objects that have several aspects. In that chapter we introduced the dis- cipline of data abstraction, according to which data structures are spec- iﬁed in terms of constructors, which create data objects, and selectors, which access the parts of compound data objects. But we now know that there is another aspect of data that Chapter 2 did not address. e desire to model systems composed of objects that have changing state leads us to the need to modify compound data objects, as well as to con- struct and select from them. In order to model compound objects with changing state, we will design data abstractions to include, in addition to selectors and constructors, operations called mutators, which mod- ify data objects. For instance, modeling a banking system requires us to change account balances. us, a data structure for representing bank accounts might admit an operation (set-balance! ⟨account⟩ ⟨new-value⟩) that changes the balance of the designated account to the designated new value. Data objects for which mutators are deﬁned are known as mutable data objects. Chapter 2 introduced pairs as a general-purpose “glue” for synthe- sizing compound data. We begin this section by deﬁning basic mutators for pairs, so that pairs can serve as building blocks for constructing mu- table data objects. ese mutators greatly enhance the representational power of pairs, enabling us to build data structures other than the se- quences and trees that we worked with in Section 2.2. We also present some examples of simulations in which complex systems are modeled as collections of objects with local state. 341 3.3.1 Mutable List Structure e basic operations on pairs—cons, car, and cdr—can be used to con- struct list structure and to select parts from list structure, but they are incapable of modifying list structure. e same is true of the list oper- ations we have used so far, such as append and list, since these can be deﬁned in terms of cons, car, and cdr. To modify list structures we need new operations. e primitive mutators for pairs are set-car! and set-cdr!. set- car! takes two arguments, the ﬁrst of which must be a pair. It modiﬁes this pair, replacing the car pointer by a pointer to the second argument of set-car!.16 As an example, suppose that x is bound to the list ((a b) c d) and y to the list (e f) as illustrated in Figure 3.12. Evaluating the expression (set-car! x y) modiﬁes the pair to which x is bound, replacing its car by the value of y. e result of the operation is shown in Figure 3.13. e structure x has been modiﬁed and would now be printed as ((e f) c d). e pairs representing the list (a b), identiﬁed by the pointer that was replaced, are now detached from the original structure.17 Compare Figure 3.13 with Figure 3.14, which illustrates the result of executing (define z (cons y (cdr x))) with x and y bound to the original lists of Figure 3.12. e variable z is now bound to a new pair created by the cons operation; the list to which x is bound is unchanged. e set-cdr! operation is similar to set-car!. e only diﬀerence is that the cdr pointer of the pair, rather than the car pointer, is replaced. e eﬀect of executing (set-cdr! x y) on the lists of Figure 3.12 is 16 set-car! and set-cdr! return implementation-dependent values. Like set!, they should be used only for their eﬀect. 17 We see from this that mutation operations on lists can create “garbage” that is not part of any accessible structure. We will see in Section 5.3.2 that Lisp memory- management systems include a garbage collector, which identiﬁes and recycles the mem- ory space used by unneeded pairs. 342 x c d a b y e f Figure 3.12: Lists x: ((a b) c d) and y: (e f). x c d a b y e f Figure 3.13: Eﬀect of (set-car! x y) on the lists in Figure 3.12. 343 x c d z a b y e f Figure 3.14: Eﬀect of (define z (cons y (cdr x))) on the lists in Figure 3.12. x c d a b y e f Figure 3.15: Eﬀect of (set-cdr! x y) on the lists in Figure 3.12. 344 shown in Figure 3.15. Here the cdr pointer of x has been replaced by the pointer to (e f). Also, the list (c d), which used to be the cdr of x, is now detached from the structure. cons builds new list structure by creating new pairs, while set-car! and set-cdr! modify existing pairs. Indeed, we could implement cons in terms of the two mutators, together with a procedure get-new-pair, which returns a new pair that is not part of any existing list structure. We obtain the new pair, set its car and cdr pointers to the designated objects, and return the new pair as the result of the cons.18 (define (cons x y) (let ((new (get-new-pair))) (set-car! new x) (set-cdr! new y) new)) Exercise 3.12: e following procedure for appending lists was introduced in Section 2.2.1: (define (append x y) (if (null? x) y (cons (car x) (append (cdr x) y)))) append forms a new list by successively consing the el- ements of x onto y. e procedure append! is similar to append, but it is a mutator rather than a constructor. It ap- pends the lists by splicing them together, modifying the ﬁ- nal pair of x so that its cdr is now y. (It is an error to call append! with an empty x.) 18 get-new-pair is one of the operations that must be implemented as part of the memory management required by a Lisp implementation. We will discuss this in Sec- tion 5.3.1. 345 (define (append! x y) (set-cdr! (last-pair x) y) x) Here last-pair is a procedure that returns the last pair in its argument: (define (last-pair x) (if (null? (cdr x)) x (last-pair (cdr x)))) Consider the interaction (define x (list 'a 'b)) (define y (list 'c 'd)) (define z (append x y)) z (a b c d) (cdr x) ⟨response⟩ (define w (append! x y)) w (a b c d) (cdr x) ⟨response⟩ What are the missing ⟨response⟩s? Draw box-and-pointer diagrams to explain your answer. Exercise 3.13: Consider the following make-cycle proce- dure, which uses the last-pair procedure deﬁned in Exer- cise 3.12: (define (make-cycle x) (set-cdr! (last-pair x) x) x) 346 Draw a box-and-pointer diagram that shows the structure z created by (define z (make-cycle (list 'a 'b 'c))) What happens if we try to compute (last-pair z)? Exercise 3.14: e following procedure is quite useful, al- though obscure: (define (mystery x) (define (loop x y) (if (null? x) y (let ((temp (cdr x))) (set-cdr! x y) (loop temp x)))) (loop x '())) loop uses the “temporary” variable temp to hold the old value of the cdr of x, since the set-cdr! on the next line destroys the cdr. Explain what mystery does in general. Suppose v is deﬁned by (define v (list 'a 'b 'c 'd)). Draw the box-and-pointer diagram that represents the list to which v is bound. Suppose that we now evalu- ate (define w (mystery v)). Draw box-and-pointer dia- grams that show the structures v and w aer evaluating this expression. What would be printed as the values of v and w? Sharing and identity We mentioned in Section 3.1.3 the theoretical issues of “sameness” and “change” raised by the introduction of assignment. ese issues arise in 347 practice when individual pairs are shared among diﬀerent data objects. For example, consider the structure formed by (define x (list 'a 'b)) (define z1 (cons x x)) As shown in Figure 3.16, z1 is a pair whose car and cdr both point to the same pair x. is sharing of x by the car and cdr of z1 is a con- sequence of the straightforward way in which cons is implemented. In general, using cons to construct lists will result in an interlinked struc- ture of pairs in which many individual pairs are shared by many diﬀer- ent structures. In contrast to Figure 3.16, Figure 3.17 shows the structure created by (define z2 (cons (list 'a 'b) (list 'a 'b))) In this structure, the pairs in the two (a b) lists are distinct, although the actual symbols are shared.19 When thought of as a list, z1 and z2 both represent “the same” list, ((a b) a b). In general, sharing is completely undetectable if we oper- ate on lists using only cons, car, and cdr. However, if we allow mutators on list structure, sharing becomes signiﬁcant. As an example of the dif- ference that sharing can make, consider the following procedure, which modiﬁes the car of the structure to which it is applied: (define (set-to-wow! x) (set-car! (car x) 'wow) x) 19 e two pairs are distinct because each call to cons returns a new pair. e symbols are shared; in Scheme there is a unique symbol with any given name. Since Scheme provides no way to mutate a symbol, this sharing is undetectable. Note also that the sharing is what enables us to compare symbols using eq?, which simply checks equality of pointers. 348 z1 x a b Figure 3.16: e list z1 formed by (cons x x). z2 a b Figure 3.17: e list z2 formed by (cons (list 'a 'b) (list 'a 'b)). Even though z1 and z2 are “the same” structure, applying set-to-wow! to them yields diﬀerent results. With z1, altering the car also changes the cdr, because in z1 the car and the cdr are the same pair. With z2, the car and cdr are distinct, so set-to-wow! modiﬁes only the car: z1 ((a b) a b) (set-to-wow! z1) ((wow b) wow b) z2 ((a b) a b) 349 (set-to-wow! z2) ((wow b) a b) One way to detect sharing in list structures is to use the predicate eq?, which we introduced in Section 2.3.1 as a way to test whether two sym- bols are equal. More generally, (eq? x y) tests whether x and y are the same object (that is, whether x and y are equal as pointers). us, with z1 and z2 as deﬁned in Figure 3.16 and Figure 3.17, (eq? (car z1) (cdr z1)) is true and (eq? (car z2) (cdr z2)) is false. As will be seen in the following sections, we can exploit sharing to greatly extend the repertoire of data structures that can be represented by pairs. On the other hand, sharing can also be dangerous, since modi- ﬁcations made to structures will also aﬀect other structures that happen to share the modiﬁed parts. e mutation operations set-car! and set- cdr! should be used with care; unless we have a good understanding of how our data objects are shared, mutation can have unanticipated re- sults.20 Exercise 3.15: Draw box-and-pointer diagrams to explain the eﬀect of set-to-wow! on the structures z1 and z2 above. Exercise 3.16: Ben Bitdiddle decides to write a procedure to count the number of pairs in any list structure. “It’s easy,” 20 e subtleties of dealing with sharing of mutable data objects reﬂect the underlying issues of “sameness” and “change” that were raised in Section 3.1.3. We mentioned there that admiing change to our language requires that a compound object must have an “identity” that is something diﬀerent from the pieces from which it is composed. In Lisp, we consider this “identity” to be the quality that is tested by eq?, i.e., by equality of pointers. Since in most Lisp implementations a pointer is essentially a memory address, we are “solving the problem” of deﬁning the identity of objects by stipulating that a data object “itself ” is the information stored in some particular set of memory locations in the computer. is suﬃces for simple Lisp programs, but is hardly a general way to resolve the issue of “sameness” in computational models. 350 he reasons. “e number of pairs in any structure is the number in the car plus the number in the cdr plus one more to count the current pair.” So Ben writes the following procedure: (define (count-pairs x) (if (not (pair? x)) 0 (+ (count-pairs (car x)) (count-pairs (cdr x)) 1))) Show that this procedure is not correct. In particular, draw box-and-pointer diagrams representing list structures made up of exactly three pairs for which Ben’s procedure would return 3; return 4; return 7; never return at all. Exercise 3.17: Devise a correct version of the count-pairs procedure of Exercise 3.16 that returns the number of dis- tinct pairs in any structure. (Hint: Traverse the structure, maintaining an auxiliary data structure that is used to keep track of which pairs have already been counted.) Exercise 3.18: Write a procedure that examines a list and determines whether it contains a cycle, that is, whether a program that tried to ﬁnd the end of the list by taking suc- cessive cdrs would go into an inﬁnite loop. Exercise 3.13 constructed such lists. Exercise 3.19: Redo Exercise 3.18 using an algorithm that takes only a constant amount of space. (is requires a very clever idea.) 351 Mutation is just assignment When we introduced compound data, we observed in Section 2.1.3 that pairs can be represented purely in terms of procedures: (define (cons x y) (define (dispatch m) (cond ((eq? m 'car) x) ((eq? m 'cdr) y) (else (error "Undefined operation: CONS" m)))) dispatch) (define (car z) (z 'car)) (define (cdr z) (z 'cdr)) e same observation is true for mutable data. We can implement mu- table data objects as procedures using assignment and local state. For instance, we can extend the above pair implementation to handle set- car! and set-cdr! in a manner analogous to the way we implemented bank accounts using make-account in Section 3.1.1: (define (cons x y) (define (set-x! v) (set! x v)) (define (set-y! v) (set! y v)) (define (dispatch m) (cond ((eq? m 'car) x) ((eq? m 'cdr) y) ((eq? m 'set-car!) set-x!) ((eq? m 'set-cdr!) set-y!) (else (error "Undefined operation: CONS" m)))) dispatch) (define (car z) (z 'car)) (define (cdr z) (z 'cdr)) (define (set-car! z new-value) ((z 'set-car!) new-value) z) 352 (define (set-cdr! z new-value) ((z 'set-cdr!) new-value) z) Assignment is all that is needed, theoretically, to account for the behav- ior of mutable data. As soon as we admit set! to our language, we raise all the issues, not only of assignment, but of mutable data in general.21 Exercise 3.20: Draw environment diagrams to illustrate the evaluation of the sequence of expressions (define x (cons 1 2)) (define z (cons x x)) (set-car! (cdr z) 17) (car x) 17 using the procedural implementation of pairs given above. (Compare Exercise 3.11.) 3.3.2 Representing eues e mutators set-car! and set-cdr! enable us to use pairs to construct data structures that cannot be built with cons, car, and cdr alone. is section shows how to use pairs to represent a data structure called a queue. Section 3.3.3 will show how to represent data structures called tables. A queue is a sequence in which items are inserted at one end (called the rear of the queue) and deleted from the other end (the front ). Fig- ure 3.18 shows an initially empty queue in which the items a and b are 21 On the other hand, from the viewpoint of implementation, assignment requires us to modify the environment, which is itself a mutable data structure. us, assignment and mutation are equipotent: Each can be implemented in terms of the other. 353 Operation Resulting Queue (define q (make-queue)) (insert-queue! q 'a) a (insert-queue! q 'b) a b (delete-queue! q) b (insert-queue! q 'c) b c (insert-queue! q 'd) b c d (delete-queue! q) c d Figure 3.18: eue operations. inserted. en a is removed, c and d are inserted, and b is removed. Be- cause items are always removed in the order in which they are inserted, a queue is sometimes called a FIFO (ﬁrst in, ﬁrst out) buﬀer. In terms of data abstraction, we can regard a queue as deﬁned by the following set of operations: • a constructor: (make-queue) returns an empty queue (a queue containing no items). • two selectors: (empty-queue? ⟨queue⟩) tests if the queue is empty. (front-queue ⟨queue⟩) returns the object at the front of the queue, signaling an error if the queue is empty; it does not modify the queue. • two mutators: (insert-queue! ⟨queue⟩ ⟨item⟩) inserts the item at the rear of the queue and returns the modiﬁed queue as its value. 354 (delete-queue! ⟨queue⟩) removes the item at the front of the queue and returns the modiﬁed queue as its value, signaling an error if the queue is empty before the deletion. Because a queue is a sequence of items, we could certainly represent it as an ordinary list; the front of the queue would be the car of the list, inserting an item in the queue would amount to appending a new ele- ment at the end of the list, and deleting an item from the queue would just be taking the cdr of the list. However, this representation is ineﬃ- cient, because in order to insert an item we must scan the list until we reach the end. Since the only method we have for scanning a list is by successive cdr operations, this scanning requires Θ(n) steps for a list of n items. A simple modiﬁcation to the list representation overcomes this disadvantage by allowing the queue operations to be implemented so that they require Θ(1) steps; that is, so that the number of steps needed is independent of the length of the queue. e diﬃculty with the list representation arises from the need to scan to ﬁnd the end of the list. e reason we need to scan is that, al- though the standard way of representing a list as a chain of pairs read- ily provides us with a pointer to the beginning of the list, it gives us no easily accessible pointer to the end. e modiﬁcation that avoids the drawback is to represent the queue as a list, together with an additional pointer that indicates the ﬁnal pair in the list. at way, when we go to insert an item, we can consult the rear pointer and so avoid scanning the list. A queue is represented, then, as a pair of pointers, front-ptr and rear-ptr, which indicate, respectively, the ﬁrst and last pairs in an or- dinary list. Since we would like the queue to be an identiﬁable object, we can use cons to combine the two pointers. us, the queue itself will be the cons of the two pointers. Figure 3.19 illustrates this representation. 355 q front-ptr rear-ptr a b c Figure 3.19: Implementation of a queue as a list with front and rear pointers. To deﬁne the queue operations we use the following procedures, which enable us to select and to modify the front and rear pointers of a queue: (define (front-ptr queue) (car queue)) (define (rear-ptr queue) (cdr queue)) (define (set-front-ptr! queue item) (set-car! queue item)) (define (set-rear-ptr! queue item) (set-cdr! queue item)) Now we can implement the actual queue operations. We will consider a queue to be empty if its front pointer is the empty list: (define (empty-queue? queue) (null? (front-ptr queue))) e make-queue constructor returns, as an initially empty queue, a pair whose car and cdr are both the empty list: (define (make-queue) (cons '() '())) 356 q front-ptr rear-ptr a b c d Figure 3.20: Result of using (insert-queue! q 'd) on the queue of Figure 3.19. To select the item at the front of the queue, we return the car of the pair indicated by the front pointer: (define (front-queue queue) (if (empty-queue? queue) (error "FRONT called with an empty queue" queue) (car (front-ptr queue)))) To insert an item in a queue, we follow the method whose result is in- dicated in Figure 3.20. We ﬁrst create a new pair whose car is the item to be inserted and whose cdr is the empty list. If the queue was initially empty, we set the front and rear pointers of the queue to this new pair. Otherwise, we modify the ﬁnal pair in the queue to point to the new pair, and also set the rear pointer to the new pair. (define (insert-queue! queue item) (let ((new-pair (cons item '()))) (cond ((empty-queue? queue) (set-front-ptr! queue new-pair) (set-rear-ptr! queue new-pair) queue) 357 q front-ptr rear-ptr a b c d Figure 3.21: Result of using (delete-queue! q) on the queue of Figure 3.20. (else (set-cdr! (rear-ptr queue) new-pair) (set-rear-ptr! queue new-pair) queue)))) To delete the item at the front of the queue, we merely modify the front pointer so that it now points at the second item in the queue, which can be found by following the cdr pointer of the ﬁrst item (see Figure 3.21):22 (define (delete-queue! queue) (cond ((empty-queue? queue) (error "DELETE! called with an empty queue" queue)) (else (set-front-ptr! queue (cdr (front-ptr queue))) queue))) 22 Ifthe ﬁrst item is the ﬁnal item in the queue, the front pointer will be the empty list aer the deletion, which will mark the queue as empty; we needn’t worry about updating the rear pointer, which will still point to the deleted item, because empty- queue? looks only at the front pointer. 358 Exercise 3.21: Ben Bitdiddle decides to test the queue im- plementation described above. He types in the procedures to the Lisp interpreter and proceeds to try them out: (define q1 (make-queue)) (insert-queue! q1 'a) ((a) a) (insert-queue! q1 'b) ((a b) b) (delete-queue! q1) ((b) b) (delete-queue! q1) (() b) “It’s all wrong!” he complains. “e interpreter’s response shows that the last item is inserted into the queue twice. And when I delete both items, the second b is still there, so the queue isn’t empty, even though it’s supposed to be.” Eva Lu Ator suggests that Ben has misunderstood what is happening. “It’s not that the items are going into the queue twice,” she explains. “It’s just that the standard Lisp printer doesn’t know how to make sense of the queue representa- tion. If you want to see the queue printed correctly, you’ll have to deﬁne your own print procedure for queues.” Ex- plain what Eva Lu is talking about. In particular, show why Ben’s examples produce the printed results that they do. Deﬁne a procedure print-queue that takes a queue as in- put and prints the sequence of items in the queue. Exercise 3.22: Instead of representing a queue as a pair of pointers, we can build a queue as a procedure with local state. e local state will consist of pointers to the begin- 359 ning and the end of an ordinary list. us, the make-queue procedure will have the form (define (make-queue) (let ((front-ptr ... ) (rear-ptr . . . )) ⟨definitions of internal procedures⟩ (define (dispatch m) . . .) dispatch)) Complete the deﬁnition of make-queue and provide imple- mentations of the queue operations using this representa- tion. Exercise 3.23: A deque (“double-ended queue”) is a sequence in which items can be inserted and deleted at either the front or the rear. Operations on deques are the constructor make-deque, the predicate empty-deque?, selectors front- deque and rear-deque, mutators front-insert-deque!, rear-insert-deque!, front-delete-deque!, and rear-delete- deque!. Show how to represent deques using pairs, and give implementations of the operations.23 All operations should be accomplished in Θ(1) steps. 3.3.3 Representing Tables When we studied various ways of representing sets in Chapter 2, we mentioned in Section 2.3.3 the task of maintaining a table of records in- dexed by identifying keys. In the implementation of data-directed pro- gramming in Section 2.4.3, we made extensive use of two-dimensional 23 Be careful not to make the interpreter try to print a structure that contains cycles. (See Exercise 3.13.) 360 table *table* a 1 b 2 c 3 Figure 3.22: A table represented as a headed list. tables, in which information is stored and retrieved using two keys. Here we see how to build tables as mutable list structures. We ﬁrst consider a one-dimensional table, in which each value is stored under a single key. We implement the table as a list of records, each of which is implemented as a pair consisting of a key and the as- sociated value. e records are glued together to form a list by pairs whose cars point to successive records. ese gluing pairs are called the backbone of the table. In order to have a place that we can change when we add a new record to the table, we build the table as a headed list. A headed list has a special backbone pair at the beginning, which holds a dummy “record”—in this case the arbitrarily chosen symbol *table*. Figure 3.22 shows the box-and-pointer diagram for the table a: 1 b: 2 c: 3 To extract information from a table we use the lookup procedure, which takes a key as argument and returns the associated value (or false if 361 there is no value stored under that key). lookup is deﬁned in terms of the assoc operation, which expects a key and a list of records as arguments. Note that assoc never sees the dummy record. assoc returns the record that has the given key as its car.24 lookup then checks to see that the resulting record returned by assoc is not false, and returns the value (the cdr) of the record. (define (lookup key table) (let ((record (assoc key (cdr table)))) (if record (cdr record) false))) (define (assoc key records) (cond ((null? records) false) ((equal? key (caar records)) (car records)) (else (assoc key (cdr records))))) To insert a value in a table under a speciﬁed key, we ﬁrst use assoc to see if there is already a record in the table with this key. If not, we form a new record by consing the key with the value, and insert this at the head of the table’s list of records, aer the dummy record. If there already is a record with this key, we set the cdr of this record to the designated new value. e header of the table provides us with a ﬁxed location to modify in order to insert the new record.25 (define (insert! key value table) (let ((record (assoc key (cdr table)))) 24 Because assoc uses equal?, it can recognize keys that are symbols, numbers, or list structure. 25 us, the ﬁrst backbone pair is the object that represents the table “itself”; that is, a pointer to the table is a pointer to this pair. is same backbone pair always starts the table. If we did not arrange things in this way, insert! would have to return a new value for the start of the table when it added a new record. 362 (if record (set-cdr! record value) (set-cdr! table (cons (cons key value) (cdr table))))) 'ok) To construct a new table, we simply create a list containing the symbol *table*: (define (make-table) (list '*table*)) Two-dimensional tables In a two-dimensional table, each value is indexed by two keys. We can construct such a table as a one-dimensional table in which each key identiﬁes a subtable. Figure 3.23 shows the box-and-pointer diagram for the table math: +: 43 letters: a: 97 -: 45 b: 98 *: 42 which has two subtables. (e subtables don’t need a special header symbol, since the key that identiﬁes the subtable serves this purpose.) When we look up an item, we use the ﬁrst key to identify the correct subtable. en we use the second key to identify the record within the subtable. (define (lookup key-1 key-2 table) (let ((subtable (assoc key-1 (cdr table)))) 363 table *table* letters a 97 b 98 math + 43 - 45 * 42 Figure 3.23: A two-dimensional table. (if subtable (let ((record (assoc key-2 (cdr subtable)))) (if record (cdr record) false)) false))) 364 To insert a new item under a pair of keys, we use assoc to see if there is a subtable stored under the ﬁrst key. If not, we build a new subtable containing the single record (key-2, value) and insert it into the table under the ﬁrst key. If a subtable already exists for the ﬁrst key, we insert the new record into this subtable, using the insertion method for one-dimensional tables described above: (define (insert! key-1 key-2 value table) (let ((subtable (assoc key-1 (cdr table)))) (if subtable (let ((record (assoc key-2 (cdr subtable)))) (if record (set-cdr! record value) (set-cdr! subtable (cons (cons key-2 value) (cdr subtable))))) (set-cdr! table (cons (list key-1 (cons key-2 value)) (cdr table))))) 'ok) Creating local tables e lookup and insert! operations deﬁned above take the table as an argument. is enables us to use programs that access more than one ta- ble. Another way to deal with multiple tables is to have separate lookup and insert! procedures for each table. We can do this by representing a table procedurally, as an object that maintains an internal table as part of its local state. When sent an appropriate message, this “table object” supplies the procedure with which to operate on the internal table. Here is a generator for two-dimensional tables represented in this fashion: 365 (define (make-table) (let ((local-table (list '*table*))) (define (lookup key-1 key-2) (let ((subtable (assoc key-1 (cdr local-table)))) (if subtable (let ((record (assoc key-2 (cdr subtable)))) (if record (cdr record) false)) false))) (define (insert! key-1 key-2 value) (let ((subtable (assoc key-1 (cdr local-table)))) (if subtable (let ((record (assoc key-2 (cdr subtable)))) (if record (set-cdr! record value) (set-cdr! subtable (cons (cons key-2 value) (cdr subtable))))) (set-cdr! local-table (cons (list key-1 (cons key-2 value)) (cdr local-table))))) 'ok) (define (dispatch m) (cond ((eq? m 'lookup-proc) lookup) ((eq? m 'insert-proc!) insert!) (else (error "Unknown operation: TABLE" m)))) dispatch)) Using make-table, we could implement the get and put operations used in Section 2.4.3 for data-directed programming, as follows: 366 (define operation-table (make-table)) (define get (operation-table 'lookup-proc)) (define put (operation-table 'insert-proc!)) get takes as arguments two keys, and put takes as arguments two keys and a value. Both operations access the same local table, which is en- capsulated within the object created by the call to make-table. Exercise 3.24: In the table implementations above, the keys are tested for equality using equal? (called by assoc). is is not always the appropriate test. For instance, we might have a table with numeric keys in which we don’t need an exact match to the number we’re looking up, but only a number within some tolerance of it. Design a table con- structor make-table that takes as an argument a same-key? procedure that will be used to test “equality” of keys. make- table should return a dispatch procedure that can be used to access appropriate lookup and insert! procedures for a local table. Exercise 3.25: Generalizing one- and two-dimensional ta- bles, show how to implement a table in which values are stored under an arbitrary number of keys and diﬀerent val- ues may be stored under diﬀerent numbers of keys. e lookup and insert! procedures should take as input a list of keys used to access the table. Exercise 3.26: To search a table as implemented above, one needs to scan through the list of records. is is basically the unordered list representation of Section 2.3.3. For large tables, it may be more eﬃcient to structure the table in a dif- ferent manner. Describe a table implementation where the 367 (key, value) records are organized using a binary tree, as- suming that keys can be ordered in some way (e.g., numer- ically or alphabetically). (Compare Exercise 2.66 of Chapter 2.) Exercise 3.27: Memoization (also called tabulation) is a tech- nique that enables a procedure to record, in a local table, values that have previously been computed. is technique can make a vast diﬀerence in the performance of a program. A memoized procedure maintains a table in which values of previous calls are stored using as keys the arguments that produced the values. When the memoized procedure is asked to compute a value, it ﬁrst checks the table to see if the value is already there and, if so, just returns that value. Otherwise, it computes the new value in the ordinary way and stores this in the table. As an example of memoization, recall from Section 1.2.2 the exponential process for com- puting Fibonacci numbers: (define (fib n) (cond ((= n 0) 0) ((= n 1) 1) (else (+ (fib (- n 1)) (fib (- n 2)))))) e memoized version of the same procedure is (define memo-fib (memoize (lambda (n) (cond ((= n 0) 0) ((= n 1) 1) (else (+ (memo-fib (- n 1)) (memo-fib (- n 2)))))))) 368 where the memoizer is deﬁned as (define (memoize f) (let ((table (make-table))) (lambda (x) (let ((previously-computed-result (lookup x table))) (or previously-computed-result (let ((result (f x))) (insert! x result table) result)))))) Draw an environment diagram to analyze the computation of (memo-fib 3). Explain why memo-fib computes the n th Fibonacci number in a number of steps proportional to n. Would the scheme still work if we had simply deﬁned memo- fib to be (memoize fib)? 3.3.4 A Simulator for Digital Circuits Designing complex digital systems, such as computers, is an important engineering activity. Digital systems are constructed by interconnect- ing simple elements. Although the behavior of these individual elements is simple, networks of them can have very complex behavior. Computer simulation of proposed circuit designs is an important tool used by digi- tal systems engineers. In this section we design a system for performing digital logic simulations. is system typiﬁes a kind of program called an event-driven simulation, in which actions (“events”) trigger further events that happen at a later time, which in turn trigger more events, and so on. Our computational model of a circuit will be composed of objects that correspond to the elementary components from which the circuit 369 Inverter And-gate Or-gate Figure 3.24: Primitive functions in the digital logic simulator. is constructed. ere are wires, which carry digital signals. A digital sig- nal may at any moment have only one of two possible values, 0 and 1. ere are also various types of digital function boxes, which connect wires carrying input signals to other output wires. Such boxes produce output signals computed from their input signals. e output signal is delayed by a time that depends on the type of the function box. For example, an inverter is a primitive function box that inverts its input. If the input signal to an inverter changes to 0, then one inverter-delay later the inverter will change its output signal to 1. If the input signal to an inverter changes to 1, then one inverter-delay later the inverter will change its output signal to 0. We draw an inverter symbolically as in Figure 3.24. An and-gate, also shown in Figure 3.24, is a primitive func- tion box with two inputs and one output. It drives its output signal to a value that is the logical and of the inputs. at is, if both of its input signals become 1, then one and-gate-delay time later the and-gate will force its output signal to be 1; otherwise the output will be 0. An or-gate is a similar two-input primitive function box that drives its output sig- nal to a value that is the logical or of the inputs. at is, the output will become 1 if at least one of the input signals is 1; otherwise the output will become 0. We can connect primitive functions together to construct more com- plex functions. To accomplish this we wire the outputs of some function boxes to the inputs of other function boxes. For example, the half-adder 370 A D S E C B Figure 3.25: A half-adder circuit. circuit shown in Figure 3.25 consists of an or-gate, two and-gates, and an inverter. It takes two input signals, A and B, and has two output sig- nals, S and C. S will become 1 whenever precisely one of A and B is 1, and C will become 1 whenever A and B are both 1. We can see from the ﬁgure that, because of the delays involved, the outputs may be gener- ated at diﬀerent times. Many of the diﬃculties in the design of digital circuits arise from this fact. We will now build a program for modeling the digital logic circuits we wish to study. e program will construct computational objects modeling the wires, which will “hold” the signals. Function boxes will be modeled by procedures that enforce the correct relationships among the signals. One basic element of our simulation will be a procedure make-wire, which constructs wires. For example, we can construct six wires as fol- lows: (define a (make-wire)) (define b (make-wire)) (define c (make-wire)) (define d (make-wire)) (define e (make-wire)) (define s (make-wire)) 371 We aach a function box to a set of wires by calling a procedure that constructs that kind of box. e arguments to the constructor procedure are the wires to be aached to the box. For example, given that we can construct and-gates, or-gates, and inverters, we can wire together the half-adder shown in Figure 3.25: (or-gate a b d) ok (and-gate a b c) ok (inverter c e) ok (and-gate d e s) ok Beer yet, we can explicitly name this operation by deﬁning a procedure half-adder that constructs this circuit, given the four external wires to be aached to the half-adder: (define (half-adder a b s c) (let ((d (make-wire)) (e (make-wire))) (or-gate a b d) (and-gate a b c) (inverter c e) (and-gate d e s) 'ok)) e advantage of making this deﬁnition is that we can use half-adder itself as a building block in creating more complex circuits. Figure 3.26, for example, shows a full-adder composed of two half-adders and an or-gate.26 We can construct a full-adder as follows: 26 A full-adder is a basic circuit element used in adding two binary numbers. Here A and B are the bits at corresponding positions in the two numbers to be added, and 372 A SUM half- adder B half- or Cout adder Cin Figure 3.26: A full-adder circuit. (define (full-adder a b c-in sum c-out) (let ((s (make-wire)) (c1 (make-wire)) (c2 (make-wire))) (half-adder b c-in s c1) (half-adder a s sum c2) (or-gate c1 c2 c-out) 'ok)) Having deﬁned full-adder as a procedure, we can now use it as a build- ing block for creating still more complex circuits. (For example, see Ex- ercise 3.30.) In essence, our simulator provides us with the tools to construct a language of circuits. If we adopt the general perspective on languages with which we approached the study of Lisp in Section 1.1, we can say that the primitive function boxes form the primitive elements of the language, that wiring boxes together provides a means of combination, and that specifying wiring paerns as procedures serves as a means of abstraction. Cin is the carry bit from the addition one place to the right. e circuit generates SUM, which is the sum bit in the corresponding position, and Cout , which is the carry bit to be propagated to the le. 373 Primitive function boxes e primitive function boxes implement the “forces” by which a change in the signal on one wire inﬂuences the signals on other wires. To build function boxes, we use the following operations on wires: • (get-signal ⟨ wire ⟩) returns the current value of the signal on the wire. • (set-signal! ⟨ wire ⟩ ⟨ new value ⟩) changes the value of the signal on the wire to the new value. • (add-action! ⟨ wire ⟩ ⟨ procedure of no arguments⟩) asserts that the designated procedure should be run whenever the signal on the wire changes value. Such procedures are the vehicles by which changes in the signal value on the wire are communicated to other wires. In addition, we will make use of a procedure after-delay that takes a time delay and a procedure to be run and executes the given procedure aer the given delay. Using these procedures, we can deﬁne the primitive digital logic functions. To connect an input to an output through an inverter, we use add-action! to associate with the input wire a procedure that will be run whenever the signal on the input wire changes value. e proce- dure computes the logical-not of the input signal, and then, aer one inverter-delay, sets the output signal to be this new value: (define (inverter input output) (define (invert-input) (let ((new-value (logical-not (get-signal input)))) 374 (after-delay inverter-delay (lambda () (set-signal! output new-value))))) (add-action! input invert-input) 'ok) (define (logical-not s) (cond ((= s 0) 1) ((= s 1) 0) (else (error "Invalid signal" s)))) An and-gate is a lile more complex. e action procedure must be run if either of the inputs to the gate changes. It computes the logical- and (using a procedure analogous to logical-not) of the values of the signals on the input wires and sets up a change to the new value to occur on the output wire aer one and-gate-delay. (define (and-gate a1 a2 output) (define (and-action-procedure) (let ((new-value (logical-and (get-signal a1) (get-signal a2)))) (after-delay and-gate-delay (lambda () (set-signal! output new-value))))) (add-action! a1 and-action-procedure) (add-action! a2 and-action-procedure) 'ok) Exercise 3.28: Deﬁne an or-gate as a primitive function box. Your or-gate constructor should be similar to and- gate. Exercise 3.29: Another way to construct an or-gate is as a compound digital logic device, built from and-gates and inverters. Deﬁne a procedure or-gate that accomplishes 375 A1 B1 C1 A2 B2 C2 A3 B3 C3 An Bn C = 0 n FA FA FA FA C S1 S2 S3 Cn-1 Sn Figure 3.27: A ripple-carry adder for n-bit numbers. this. What is the delay time of the or-gate in terms of and- gate-delay and inverter-delay? Exercise 3.30: Figure 3.27 shows a ripple-carry adder formed by stringing together n full-adders. is is the simplest form of parallel adder for adding two n-bit binary numbers. e inputs A 1 , A 2 , A 3 , . . ., An and B 1 , B 2 , B 3 , . . ., Bn are the two binary numbers to be added (each A k and B k is a 0 or a 1). e circuit generates S 1 , S 2 , S 3 , . . ., Sn , the n bits of the sum, and C, the carry from the addition. Write a proce- dure ripple-carry-adder that generates this circuit. e procedure should take as arguments three lists of n wires each—the A k , the B k , and the S k —and also another wire C. e major drawback of the ripple-carry adder is the need to wait for the carry signals to propagate. What is the delay needed to obtain the complete output from an n-bit ripple- carry adder, expressed in terms of the delays for and-gates, or-gates, and inverters? 376 Representing wires A wire in our simulation will be a computational object with two local state variables: a signal-value (initially taken to be 0) and a collec- tion of action-procedures to be run when the signal changes value. We implement the wire, using message-passing style, as a collection of local procedures together with a dispatch procedure that selects the ap- propriate local operation, just as we did with the simple bank-account object in Section 3.1.1: (define (make-wire) (let ((signal-value 0) (action-procedures '())) (define (set-my-signal! new-value) (if (not (= signal-value new-value)) (begin (set! signal-value new-value) (call-each action-procedures)) 'done)) (define (accept-action-procedure! proc) (set! action-procedures (cons proc action-procedures)) (proc)) (define (dispatch m) (cond ((eq? m 'get-signal) signal-value) ((eq? m 'set-signal!) set-my-signal!) ((eq? m 'add-action!) accept-action-procedure!) (else (error "Unknown operation: WIRE" m)))) dispatch)) e local procedure set-my-signal! tests whether the new signal value changes the signal on the wire. If so, it runs each of the action proce- dures, using the following procedure call-each, which calls each of the items in a list of no-argument procedures: (define (call-each procedures) 377 (if (null? procedures) 'done (begin ((car procedures)) (call-each (cdr procedures))))) e local procedure accept-action-procedure! adds the given proce- dure to the list of procedures to be run, and then runs the new procedure once. (See Exercise 3.31.) With the local dispatch procedure set up as speciﬁed, we can pro- vide the following procedures to access the local operations on wires:27 (define (get-signal wire) (wire 'get-signal)) (define (set-signal! wire new-value) ((wire 'set-signal!) new-value)) (define (add-action! wire action-procedure) ((wire 'add-action!) action-procedure)) Wires, which have time-varying signals and may be incrementally at- tached to devices, are typical of mutable objects. We have modeled them as procedures with local state variables that are modiﬁed by assignment. When a new wire is created, a new set of state variables is allocated (by the let expression in make-wire) and a new dispatch procedure is constructed and returned, capturing the environment with the new state variables. 27 ese procedures are simply syntactic sugar that allow us to use ordinary pro- cedural syntax to access the local procedures of objects. It is striking that we can in- terchange the role of “procedures” and “data” in such a simple way. For example, if we write (wire 'get-signal) we think of wire as a procedure that is called with the mes- sage get-signal as input. Alternatively, writing (get-signal wire) encourages us to think of wire as a data object that is the input to a procedure get-signal. e truth of the maer is that, in a language in which we can deal with procedures as objects, there is no fundamental diﬀerence between “procedures” and “data,” and we can choose our syntactic sugar to allow us to program in whatever style we choose. 378 e wires are shared among the various devices that have been con- nected to them. us, a change made by an interaction with one device will aﬀect all the other devices aached to the wire. e wire communi- cates the change to its neighbors by calling the action procedures pro- vided to it when the connections were established. The agenda e only thing needed to complete the simulator is after-delay. e idea here is that we maintain a data structure, called an agenda, that contains a schedule of things to do. e following operations are deﬁned for agendas: • (make-agenda) returns a new empty agenda. • (empty-agenda? ⟨ agenda ⟩) is true if the speciﬁed agenda is empty. • (first-agenda-item ⟨ agenda ⟩) returns the ﬁrst item on the agenda. • (remove-first-agenda-item! ⟨ agenda ⟩) modiﬁes the agenda by removing the ﬁrst item. • (add-to-agenda! ⟨ time ⟩ ⟨ action ⟩ ⟨ agenda ⟩) modiﬁes the agenda by adding the given action procedure to be run at the spec- iﬁed time. • (current-time ⟨ agenda ⟩) returns the current simulation time. e particular agenda that we use is denoted by the-agenda. e pro- cedure after-delay adds new elements to the-agenda: 379 (define (after-delay delay action) (add-to-agenda! (+ delay (current-time the-agenda)) action the-agenda)) e simulation is driven by the procedure propagate, which operates on the-agenda, executing each procedure on the agenda in sequence. In general, as the simulation runs, new items will be added to the agenda, and propagate will continue the simulation as long as there are items on the agenda: (define (propagate) (if (empty-agenda? the-agenda) 'done (let ((first-item (first-agenda-item the-agenda))) (first-item) (remove-first-agenda-item! the-agenda) (propagate)))) A sample simulation e following procedure, which places a “probe” on a wire, shows the simulator in action. e probe tells the wire that, whenever its signal changes value, it should print the new signal value, together with the current time and a name that identiﬁes the wire: (define (probe name wire) (add-action! wire (lambda () (newline) (display name) (display " ") (display (current-time the-agenda)) (display " New-value = ") (display (get-signal wire))))) 380 We begin by initializing the agenda and specifying delays for the prim- itive function boxes: (define the-agenda (make-agenda)) (define inverter-delay 2) (define and-gate-delay 3) (define or-gate-delay 5) Now we deﬁne four wires, placing probes on two of them: (define input-1 (make-wire)) (define input-2 (make-wire)) (define sum (make-wire)) (define carry (make-wire)) (probe 'sum sum) sum 0 New-value = 0 (probe 'carry carry) carry 0 New-value = 0 Next we connect the wires in a half-adder circuit (as in Figure 3.25), set the signal on input-1 to 1, and run the simulation: (half-adder input-1 input-2 sum carry) ok (set-signal! input-1 1) done (propagate) sum 8 New-value = 1 done e sum signal changes to 1 at time 8. We are now eight time units from the beginning of the simulation. At this point, we can set the signal on input-2 to 1 and allow the values to propagate: 381 (set-signal! input-2 1) done (propagate) carry 11 New-value = 1 sum 16 New-value = 0 done e carry changes to 1 at time 11 and the sum changes to 0 at time 16. Exercise 3.31: e internal procedure accept-action-procedure! deﬁned in make-wire speciﬁes that when a new action pro- cedure is added to a wire, the procedure is immediately run. Explain why this initialization is necessary. In particu- lar, trace through the half-adder example in the paragraphs above and say how the system’s response would diﬀer if we had deﬁned accept-action-procedure! as (define (accept-action-procedure! proc) (set! action-procedures (cons proc action-procedures))) Implementing the agenda Finally, we give details of the agenda data structure, which holds the procedures that are scheduled for future execution. e agenda is made up of time segments. Each time segment is a pair consisting of a number (the time) and a queue (see Exercise 3.32) that holds the procedures that are scheduled to be run during that time segment. (define (make-time-segment time queue) (cons time queue)) 382 (define (segment-time s) (car s)) (define (segment-queue s) (cdr s)) We will operate on the time-segment queues using the queue operations described in Section 3.3.2. e agenda itself is a one-dimensional table of time segments. It diﬀers from the tables described in Section 3.3.3 in that the segments will be sorted in order of increasing time. In addition, we store the current time (i.e., the time of the last action that was processed) at the head of the agenda. A newly constructed agenda has no time segments and has a current time of 0:28 (define (make-agenda) (list 0)) (define (current-time agenda) (car agenda)) (define (set-current-time! agenda time) (set-car! agenda time)) (define (segments agenda) (cdr agenda)) (define (set-segments! agenda segments) (set-cdr! agenda segments)) (define (first-segment agenda) (car (segments agenda))) (define (rest-segments agenda) (cdr (segments agenda))) An agenda is empty if it has no time segments: (define (empty-agenda? agenda) (null? (segments agenda))) To add an action to an agenda, we ﬁrst check if the agenda is empty. If so, we create a time segment for the action and install this in the agenda. Otherwise, we scan the agenda, examining the time of each segment. If we ﬁnd a segment for our appointed time, we add the action to the 28 e agenda is a headed list, like the tables in Section 3.3.3, but since the list is headed by the time, we do not need an additional dummy header (such as the *table* symbol used with tables). 383 associated queue. If we reach a time later than the one to which we are appointed, we insert a new time segment into the agenda just before it. If we reach the end of the agenda, we must create a new time segment at the end. (define (add-to-agenda! time action agenda) (define (belongs-before? segments) (or (null? segments) (< time (segment-time (car segments))))) (define (make-new-time-segment time action) (let ((q (make-queue))) (insert-queue! q action) (make-time-segment time q))) (define (add-to-segments! segments) (if (= (segment-time (car segments)) time) (insert-queue! (segment-queue (car segments)) action) (let ((rest (cdr segments))) (if (belongs-before? rest) (set-cdr! segments (cons (make-new-time-segment time action) (cdr segments))) (add-to-segments! rest))))) (let ((segments (segments agenda))) (if (belongs-before? segments) (set-segments! agenda (cons (make-new-time-segment time action) segments)) (add-to-segments! segments)))) e procedure that removes the ﬁrst item from the agenda deletes the item at the front of the queue in the ﬁrst time segment. If this deletion 384 makes the time segment empty, we remove it from the list of segments:29 (define (remove-first-agenda-item! agenda) (let ((q (segment-queue (first-segment agenda)))) (delete-queue! q) (if (empty-queue? q) (set-segments! agenda (rest-segments agenda))))) e ﬁrst agenda item is found at the head of the queue in the ﬁrst time segment. Whenever we extract an item, we also update the cur- rent time:30 (define (first-agenda-item agenda) (if (empty-agenda? agenda) (error "Agenda is empty: FIRST-AGENDA-ITEM") (let ((first-seg (first-segment agenda))) (set-current-time! agenda (segment-time first-seg)) (front-queue (segment-queue first-seg))))) Exercise 3.32: e procedures to be run during each time segment of the agenda are kept in a queue. us, the pro- cedures for each segment are called in the order in which they were added to the agenda (ﬁrst in, ﬁrst out). Explain why this order must be used. In particular, trace the behav- ior of an and-gate whose inputs change from 0, 1 to 1, 0 29 Observe that the if expression in this procedure has no ⟨alternative ⟩ expression. Such a “one-armed if statement” is used to decide whether to do something, rather than to select between two expressions. An if expression returns an unspeciﬁed value if the predicate is false and there is no ⟨alternative ⟩. 30 In this way, the current time will always be the time of the action most recently processed. Storing this time at the head of the agenda ensures that it will still be avail- able even if the associated time segment has been deleted. 385 in the same segment and say how the behavior would dif- fer if we stored a segment’s procedures in an ordinary list, adding and removing procedures only at the front (last in, ﬁrst out). 3.3.5 Propagation of Constraints Computer programs are traditionally organized as one-directional com- putations, which perform operations on prespeciﬁed arguments to pro- duce desired outputs. On the other hand, we oen model systems in terms of relations among quantities. For example, a mathematical model of a mechanical structure might include the information that the deﬂec- tion d of a metal rod is related to the force F on the rod, the length L of the rod, the cross-sectional area A, and the elastic modulus E via the equation dAE = F L. Such an equation is not one-directional. Given any four of the quanti- ties, we can use it to compute the ﬁh. Yet translating the equation into a traditional computer language would force us to choose one of the quantities to be computed in terms of the other four. us, a procedure for computing the area A could not be used to compute the deﬂection d, even though the computations of A and d arise from the same equa- tion.31 31 Constraint propagation ﬁrst appeared in the incredibly forward-looking system of Ivan Sutherland (1963). A beautiful constraint-propagation system based on the Smalltalk language was developed by Alan Borning (1977) at Xerox Palo Alto Research Center. Sussman, Stallman, and Steele applied constraint propagation to elec- trical circuit analysis (Sussman and Stallman 1975; Sussman and Steele 1980). TK!Solver (Konopasek and Jayaraman 1984) is an extensive modeling environment based on constraints. 386 In this section, we sketch the design of a language that enables us to work in terms of relations themselves. e primitive elements of the language are primitive constraints, which state that certain relations hold between quantities. For example, (adder a b c) speciﬁes that the quan- tities a, b, and c must be related by the equation a + b = c, (multiplier x y z) expresses the constraint xy = z, and (constant 3.14 x) says that the value of x must be 3.14. Our language provides a means of combining primitive constraints in order to express more complex relations. We combine constraints by constructing constraint networks, in which constraints are joined by connectors. A connector is an object that “holds” a value that may par- ticipate in one or more constraints. For example, we know that the re- lationship between Fahrenheit and Celsius temperatures is 9C = 5(F − 32). Such a constraint can be thought of as a network consisting of primitive adder, multiplier, and constant constraints (Figure 3.28). In the ﬁgure, we see on the le a multiplier box with three terminals, labeled m1, m2, and p. ese connect the multiplier to the rest of the network as follows: e m1 terminal is linked to a connector C, which will hold the Celsius temperature. e m2 terminal is linked to a connector w, which is also linked to a constant box that holds 9. e p terminal, which the multiplier box constrains to be the product of m1 and m2, is linked to the p terminal of another multiplier box, whose m2 is connected to a constant 5 and whose m1 is connected to one of the terms in a sum. Computation by such a network proceeds as follows: When a con- nector is given a value (by the user or by a constraint box to which it is linked), it awakens all of its associated constraints (except for the constraint that just awakened it) to inform them that it has a value. 387 v C m1 m1 a1 u * p p * + s F m2 m2 a2 w x y 9 5 32 Figure 3.28: e relation 9C = 5(F − 32) expressed as a constraint network. Each awakened constraint box then polls its connectors to see if there is enough information to determine a value for a connector. If so, the box sets that connector, which then awakens all of its associated con- straints, and so on. For instance, in conversion between Celsius and Fahrenheit, w, x , and y are immediately set by the constant boxes to 9, 5, and 32, respectively. e connectors awaken the multipliers and the adder, which determine that there is not enough information to pro- ceed. If the user (or some other part of the network) sets C to a value (say 25), the lemost multiplier will be awakened, and it will set u to 25 · 9 = 225. en u awakens the second multiplier, which sets v to 45, and v awakens the adder, which sets f to 77. Using the constraint system To use the constraint system to carry out the temperature computation outlined above, we ﬁrst create two connectors, C and F, by calling the constructor make-connector, and link C and F in an appropriate net- work: (define C (make-connector)) (define F (make-connector)) 388 (celsius-fahrenheit-converter C F) ok e procedure that creates the network is deﬁned as follows: (define (celsius-fahrenheit-converter c f) (let ((u (make-connector)) (v (make-connector)) (w (make-connector)) (x (make-connector)) (y (make-connector))) (multiplier c w u) (multiplier v x u) (adder v y f) (constant 9 w) (constant 5 x) (constant 32 y) 'ok)) is procedure creates the internal connectors u, v, w, x, and y, and links them as shown in Figure 3.28 using the primitive constraint construc- tors adder, multiplier, and constant. Just as with the digital-circuit simulator of Section 3.3.4, expressing these combinations of primitive elements in terms of procedures automatically provides our language with a means of abstraction for compound objects. To watch the network in action, we can place probes on the con- nectors C and F, using a probe procedure similar to the one we used to monitor wires in Section 3.3.4. Placing a probe on a connector will cause a message to be printed whenever the connector is given a value: (probe "Celsius temp" C) (probe "Fahrenheit temp" F) Next we set the value of C to 25. (e third argument to set-value! tells C that this directive comes from the user.) 389 (set-value! C 25 'user) Probe: Celsius temp = 25 Probe: Fahrenheit temp = 77 done e probe on C awakens and reports the value. C also propagates its value through the network as described above. is sets F to 77, which is reported by the probe on F. Now we can try to set F to a new value, say 212: (set-value! F 212 'user) Error! Contradiction (77 212) e connector complains that it has sensed a contradiction: Its value is 77, and someone is trying to set it to 212. If we really want to reuse the network with new values, we can tell C to forget its old value: (forget-value! C 'user) Probe: Celsius temp = ? Probe: Fahrenheit temp = ? done C ﬁnds that the user, who set its value originally, is now retracting that value, so C agrees to lose its value, as shown by the probe, and informs the rest of the network of this fact. is information eventually prop- agates to F, which now ﬁnds that it has no reason for continuing to believe that its own value is 77. us, F also gives up its value, as shown by the probe. Now that F has no value, we are free to set it to 212: (set-value! F 212 'user) Probe: Fahrenheit temp = 212 Probe: Celsius temp = 100 done 390 is new value, when propagated through the network, forces C to have a value of 100, and this is registered by the probe on C. Notice that the very same network is being used to compute C given F and to compute F given C. is nondirectionality of computation is the distinguishing feature of constraint-based systems. Implementing the constraint system e constraint system is implemented via procedural objects with local state, in a manner very similar to the digital-circuit simulator of Sec- tion 3.3.4. Although the primitive objects of the constraint system are somewhat more complex, the overall system is simpler, since there is no concern about agendas and logic delays. e basic operations on connectors are the following: • (has-value? ⟨connector⟩) tells whether the connector has a value. • (get-value ⟨connector⟩) returns the connector’s current value. • (set-value! ⟨connector⟩ ⟨new-value⟩ ⟨informant⟩) indicates that the informant is requesting the connector to set its value to the new value. • (forget-value! ⟨connector⟩ ⟨retractor⟩) tells the connector that the retractor is requesting it to forget its value. • (connect ⟨connector⟩ ⟨new-constraint⟩) tells the connector to participate in the new constraint. e connectors communicate with the constraints by means of the pro- cedures inform-about-value, which tells the given constraint that the 391 connector has a value, and inform-about-no-value, which tells the constraint that the connector has lost its value. adder constructs an adder constraint among summand connectors a1 and a2 and a sum connector. An adder is implemented as a procedure with local state (the procedure me below): (define (adder a1 a2 sum) (define (process-new-value) (cond ((and (has-value? a1) (has-value? a2)) (set-value! sum (+ (get-value a1) (get-value a2)) me)) ((and (has-value? a1) (has-value? sum)) (set-value! a2 (- (get-value sum) (get-value a1)) me)) ((and (has-value? a2) (has-value? sum)) (set-value! a1 (- (get-value sum) (get-value a2)) me)))) (define (process-forget-value) (forget-value! sum me) (forget-value! a1 me) (forget-value! a2 me) (process-new-value)) (define (me request) (cond ((eq? request 'I-have-a-value) (process-new-value)) ((eq? request 'I-lost-my-value) (process-forget-value)) (else (error "Unknown request: ADDER" request)))) (connect a1 me) (connect a2 me) (connect sum me) me) 392 adder connects the new adder to the designated connectors and returns it as its value. e procedure me, which represents the adder, acts as a dispatch to the local procedures. e following “syntax interfaces” (see Footnote 27 in Section 3.3.4) are used in conjunction with the dispatch: (define (inform-about-value constraint) (constraint 'I-have-a-value)) (define (inform-about-no-value constraint) (constraint 'I-lost-my-value)) e adder’s local procedure process-new-value is called when the adder is informed that one of its connectors has a value. e adder ﬁrst checks to see if both a1 and a2 have values. If so, it tells sum to set its value to the sum of the two addends. e informant argument to set-value! is me, which is the adder object itself. If a1 and a2 do not both have values, then the adder checks to see if perhaps a1 and sum have values. If so, it sets a2 to the diﬀerence of these two. Finally, if a2 and sum have values, this gives the adder enough information to set a1. If the adder is told that one of its connectors has lost a value, it requests that all of its con- nectors now lose their values. (Only those values that were set by this adder are actually lost.) en it runs process-new-value. e reason for this last step is that one or more connectors may still have a value (that is, a connector may have had a value that was not originally set by the adder), and these values may need to be propagated back through the adder. A multiplier is very similar to an adder. It will set its product to 0 if either of the factors is 0, even if the other factor is not known. (define (multiplier m1 m2 product) (define (process-new-value) (cond ((or (and (has-value? m1) (= (get-value m1) 0)) (and (has-value? m2) (= (get-value m2) 0))) 393 (set-value! product 0 me)) ((and (has-value? m1) (has-value? m2)) (set-value! product (* (get-value m1) (get-value m2)) me)) ((and (has-value? product) (has-value? m1)) (set-value! m2 (/ (get-value product) (get-value m1)) me)) ((and (has-value? product) (has-value? m2)) (set-value! m1 (/ (get-value product) (get-value m2)) me)))) (define (process-forget-value) (forget-value! product me) (forget-value! m1 me) (forget-value! m2 me) (process-new-value)) (define (me request) (cond ((eq? request 'I-have-a-value) (process-new-value)) ((eq? request 'I-lost-my-value) (process-forget-value)) (else (error "Unknown request: MULTIPLIER" request)))) (connect m1 me) (connect m2 me) (connect product me) me) A constant constructor simply sets the value of the designated con- nector. Any I-have-a-value or I-lost-my-value message sent to the constant box will produce an error. 394 (define (constant value connector) (define (me request) (error "Unknown request: CONSTANT" request)) (connect connector me) (set-value! connector value me) me) Finally, a probe prints a message about the seing or unseing of the designated connector: (define (probe name connector) (define (print-probe value) (newline) (display "Probe: ") (display name) (display " = ") (display value)) (define (process-new-value) (print-probe (get-value connector))) (define (process-forget-value) (print-probe "?")) (define (me request) (cond ((eq? request 'I-have-a-value) (process-new-value)) ((eq? request 'I-lost-my-value) (process-forget-value)) (else (error "Unknown request: PROBE" request)))) (connect connector me) me) Representing connectors A connector is represented as a procedural object with local state vari- ables value, the current value of the connector; informant, the object that set the connector’s value; and constraints, a list of the constraints in which the connector participates. (define (make-connector) (let ((value false) (informant false) (constraints '())) (define (set-my-value newval setter) 395 (cond ((not (has-value? me)) (set! value newval) (set! informant setter) (for-each-except setter inform-about-value constraints)) ((not (= value newval)) (error "Contradiction" (list value newval))) (else 'ignored))) (define (forget-my-value retractor) (if (eq? retractor informant) (begin (set! informant false) (for-each-except retractor inform-about-no-value constraints)) 'ignored)) (define (connect new-constraint) (if (not (memq new-constraint constraints)) (set! constraints (cons new-constraint constraints))) (if (has-value? me) (inform-about-value new-constraint)) 'done) (define (me request) (cond ((eq? request 'has-value?) (if informant true false)) ((eq? request 'value) value) ((eq? request 'set-value!) set-my-value) ((eq? request 'forget) forget-my-value) ((eq? request 'connect) connect) (else (error "Unknown operation: CONNECTOR" request)))) me)) 396 e connector’s local procedure set-my-value is called when there is a request to set the connector’s value. If the connector does not cur- rently have a value, it will set its value and remember as informant the constraint that requested the value to be set.32 en the connector will notify all of its participating constraints except the constraint that requested the value to be set. is is accomplished using the follow- ing iterator, which applies a designated procedure to all items in a list except a given one: (define (for-each-except exception procedure list) (define (loop items) (cond ((null? items) 'done) ((eq? (car items) exception) (loop (cdr items))) (else (procedure (car items)) (loop (cdr items))))) (loop list)) If a connector is asked to forget its value, it runs the local procedure forget-my-value, which ﬁrst checks to make sure that the request is coming from the same object that set the value originally. If so, the con- nector informs its associated constraints about the loss of the value. e local procedure connect adds the designated new constraint to the list of constraints if it is not already in that list. en, if the connector has a value, it informs the new constraint of this fact. e connector’s procedure me serves as a dispatch to the other in- ternal procedures and also represents the connector as an object. e following procedures provide a syntax interface for the dispatch: (define (has-value? connector) (connector 'has-value?)) 32 e setter might not be a constraint. In our temperature example, we used user as the setter. 397 (define (get-value connector) (connector 'value)) (define (set-value! connector new-value informant) ((connector 'set-value!) new-value informant)) (define (forget-value! connector retractor) ((connector 'forget) retractor)) (define (connect connector new-constraint) ((connector 'connect) new-constraint)) Exercise 3.33: Using primitive multiplier, adder, and con- stant constraints, deﬁne a procedure averager that takes three connectors a, b, and c as inputs and establishes the constraint that the value of c is the average of the values of a and b. Exercise 3.34: Louis Reasoner wants to build a squarer, a constraint device with two terminals such that the value of connector b on the second terminal will always be the square of the value a on the ﬁrst terminal. He proposes the following simple device made from a multiplier: (define (squarer a b) (multiplier a a b)) ere is a serious ﬂaw in this idea. Explain. Exercise 3.35: Ben Bitdiddle tells Louis that one way to avoid the trouble in Exercise 3.34 is to deﬁne a squarer as a new primitive constraint. Fill in the missing portions in Ben’s outline for a procedure to implement such a con- straint: 398 (define (squarer a b) (define (process-new-value) (if (has-value? b) (if (< (get-value b) 0) (error "square less than 0: SQUARER" (get-value b)) ⟨alternative1⟩) ⟨alternative2⟩)) (define (process-forget-value) ⟨body1⟩) (define (me request) ⟨body2⟩) ⟨rest of definition⟩ me) Exercise 3.36: Suppose we evaluate the following sequence of expressions in the global environment: (define a (make-connector)) (define b (make-connector)) (set-value! a 10 'user) At some time during evaluation of the set-value!, the fol- lowing expression from the connector’s local procedure is evaluated: (for-each-except setter inform-about-value constraints) Draw an environment diagram showing the environment in which the above expression is evaluated. Exercise 3.37: e celsius-fahrenheit-converter pro- cedure is cumbersome when compared with a more expression- oriented style of deﬁnition, such as 399 (define (celsius-fahrenheit-converter x) (c+ (c* (c/ (cv 9) (cv 5)) x) (cv 32))) (define C (make-connector)) (define F (celsius-fahrenheit-converter C)) Here c+, c*, etc. are the “constraint” versions of the arith- metic operations. For example, c+ takes two connectors as arguments and returns a connector that is related to these by an adder constraint: (define (c+ x y) (let ((z (make-connector))) (adder x y z) z)) Deﬁne analogous procedures c-, c*, c/, and cv (constant value) that enable us to deﬁne compound constraints as in the converter example above.33 33 e expression-oriented format is convenient because it avoids the need to name the intermediate expressions in a computation. Our original formulation of the con- straint language is cumbersome in the same way that many languages are cumbersome when dealing with operations on compound data. For example, if we wanted to com- pute the product (a +b) · (c +d), where the variables represent vectors, we could work in “imperative style,” using procedures that set the values of designated vector arguments but do not themselves return vectors as values: (v-sum a b temp1) (v-sum c d temp2) (v-prod temp1 temp2 answer) Alternatively, we could deal with expressions, using procedures that return vectors as values, and thus avoid explicitly mentioning temp1 and temp2: (define answer (v-prod (v-sum a b) (v-sum c d))) 400 3.4 Concurrency: Time Is of the Essence We’ve seen the power of computational objects with local state as tools for modeling. Yet, as Section 3.1.3 warned, this power extracts a price: the loss of referential transparency, giving rise to a thicket of questions about sameness and change, and the need to abandon the substitution model of evaluation in favor of the more intricate environment model. e central issue lurking beneath the complexity of state, sameness, and change is that by introducing assignment we are forced to admit time into our computational models. Before we introduced assignment, all our programs were timeless, in the sense that any expression that has a value always has the same value. In contrast, recall the example of modeling withdrawals from a bank account and returning the resulting balance, introduced at the beginning of Section 3.1.1: (withdraw 25) 75 (withdraw 25) 50 Since Lisp allows us to return compound objects as values of procedures, we can trans- form our imperative-style constraint language into an expression-oriented style as shown in this exercise. In languages that are impoverished in handling compound ob- jects, such as Algol, Basic, and Pascal (unless one explicitly uses Pascal pointer vari- ables), one is usually stuck with the imperative style when manipulating compound objects. Given the advantage of the expression-oriented format, one might ask if there is any reason to have implemented the system in imperative style, as we did in this section. One reason is that the non-expression-oriented constraint language provides a handle on constraint objects (e.g., the value of the adder procedure) as well as on connector objects. is is useful if we wish to extend the system with new operations that communicate with constraints directly rather than only indirectly via operations on connectors. Although it is easy to implement the expression-oriented style in terms of the imperative implementation, it is very diﬃcult to do the converse. 401 Here successive evaluations of the same expression yield diﬀerent val- ues. is behavior arises from the fact that the execution of assignment statements (in this case, assignments to the variable balance) delineates moments in time when values change. e result of evaluating an ex- pression depends not only on the expression itself, but also on whether the evaluation occurs before or aer these moments. Building models in terms of computational objects with local state forces us to confront time as an essential concept in programming. We can go further in structuring computational models to match our perception of the physical world. Objects in the world do not change one at a time in sequence. Rather we perceive them as acting concur- rently —all at once. So it is oen natural to model systems as collections of computational processes that execute concurrently. Just as we can make our programs modular by organizing models in terms of objects with separate local state, it is oen appropriate to divide computational models into parts that evolve separately and concurrently. Even if the programs are to be executed on a sequential computer, the practice of writing programs as if they were to be executed concurrently forces the programmer to avoid inessential timing constraints and thus makes programs more modular. In addition to making programs more modular, concurrent compu- tation can provide a speed advantage over sequential computation. Se- quential computers execute only one operation at a time, so the amount of time it takes to perform a task is proportional to the total number of operations performed.34 However, if it is possible to decompose a 34 Most real processors actually execute a few operations at a time, following a strat- egy called pipelining. Although this technique greatly improves the eﬀective utilization of the hardware, it is used only to speed up the execution of a sequential instruction stream, while retaining the behavior of the sequential program. 402 problem into pieces that are relatively independent and need to com- municate only rarely, it may be possible to allocate pieces to separate computing processors, producing a speed advantage proportional to the number of processors available. Unfortunately, the complexities introduced by assignment become even more problematic in the presence of concurrency. e fact of con- current execution, either because the world operates in parallel or be- cause our computers do, entails additional complexity in our under- standing of time. 3.4.1 The Nature of Time in Concurrent Systems On the surface, time seems straightforward. It is an ordering imposed on events.35 For any events A and B, either A occurs before B, A and B are simultaneous, or A occurs aer B. For instance, returning to the bank account example, suppose that Peter withdraws $10 and Paul with- draws $25 from a joint account that initially contains $100, leaving $65 in the account. Depending on the order of the two withdrawals, the sequence of balances in the account is either $100 → $90 → $65 or $100 → $75 → $65 . In a computer implementation of the banking sys- tem, this changing sequence of balances could be modeled by successive assignments to a variable balance. In complex situations, however, such a view can be problematic. Suppose that Peter and Paul, and other people besides, are accessing the same bank account through a network of banking machines distributed all over the world. e actual sequence of balances in the account will depend critically on the detailed timing of the accesses and the details of the communication among the machines. 35 Toquote some graﬃti seen on a Cambridge building wall: “Time is a device that was invented to keep everything from happening at once.” 403 is indeterminacy in the order of events can pose serious prob- lems in the design of concurrent systems. For instance, suppose that the withdrawals made by Peter and Paul are implemented as two separate processes sharing a common variable balance, each process speciﬁed by the procedure given in Section 3.1.1: (define (withdraw amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")) If the two processes operate independently, then Peter might test the balance and aempt to withdraw a legitimate amount. However, Paul might withdraw some funds in between the time that Peter checks the balance and the time Peter completes the withdrawal, thus invalidating Peter’s test. ings can be worse still. Consider the expression (set! balance (- balance amount)) executed as part of each withdrawal process. is consists of three steps: (1) accessing the value of the balance variable; (2) computing the new balance; (3) seing balance to this new value. If Peter and Paul’s with- drawals execute this statement concurrently, then the two withdrawals might interleave the order in which they access balance and set it to the new value. e timing diagram in Figure 3.29 depicts an order of events where balance starts at 100, Peter withdraws 10, Paul withdraws 25, and yet the ﬁnal value of balance is 75. As shown in the diagram, the reason for this anomaly is that Paul’s assignment of 75 to balance is made under the assumption that the value of balance to be decremented is 100. at 404 Peter Bank Paul $100 Access balance: $100 Access balance: $100 new value: 100 –10 = 90 new value: 100 – 25 = 75 set! balance to $90 $90 set! balance to $75 $75 time Figure 3.29: Timing diagram showing how interleaving the order of events in two banking withdrawals can lead to an incorrect ﬁnal balance. 405 assumption, however, became invalid when Peter changed balance to 90. is is a catastrophic failure for the banking system, because the total amount of money in the system is not conserved. Before the trans- actions, the total amount of money was $100. Aerwards, Peter has $10, Paul has $25, and the bank has $75.36 e general phenomenon illustrated here is that several processes may share a common state variable. What makes this complicated is that more than one process may be trying to manipulate the shared state at the same time. For the bank account example, during each transaction, each customer should be able to act as if the other customers did not exist. When a customer changes the balance in a way that depends on the balance, he must be able to assume that, just before the moment of change, the balance is still what he thought it was. Correct behavior of concurrent programs e above example typiﬁes the subtle bugs that can creep into concur- rent programs. e root of this complexity lies in the assignments to variables that are shared among the diﬀerent processes. We already know that we must be careful in writing programs that use set!, be- cause the results of a computation depend on the order in which the 36 An even worse failure for this system could occur if the two set! operations at- tempt to change the balance simultaneously, in which case the actual data appearing in memory might end up being a random combination of the information being writ- ten by the two processes. Most computers have interlocks on the primitive memory- write operations, which protect against such simultaneous access. Even this seemingly simple kind of protection, however, raises implementation challenges in the design of multiprocessing computers, where elaborate cache-coherence protocols are required to ensure that the various processors will maintain a consistent view of memory contents, despite the fact that data may be replicated (“cached”) among the diﬀerent processors to increase the speed of memory access. 406 assignments occur.37 With concurrent processes we must be especially careful about assignments, because we may not be able to control the order of the assignments made by the diﬀerent processes. If several such changes might be made concurrently (as with two depositors accessing a joint account) we need some way to ensure that our system behaves correctly. For example, in the case of withdrawals from a joint bank ac- count, we must ensure that money is conserved. To make concurrent programs behave correctly, we may have to place some restrictions on concurrent execution. One possible restriction on concurrency would stipulate that no two operations that change any shared state variables can occur at the same time. is is an extremely stringent requirement. For distributed bank- ing, it would require the system designer to ensure that only one trans- action could proceed at a time. is would be both ineﬃcient and overly conservative. Figure 3.30 shows Peter and Paul sharing a bank account, where Paul has a private account as well. e diagram illustrates two withdrawals from the shared account (one by Peter and one by Paul) and a deposit to Paul’s private account.38 e two withdrawals from the shared account must not be concurrent (since both access and up- date the same account), and Paul’s deposit and withdrawal must not be concurrent (since both access and update the amount in Paul’s wallet). But there should be no problem permiing Paul’s deposit to his pri- vate account to proceed concurrently with Peter’s withdrawal from the shared account. A less stringent restriction on concurrency would ensure that a con- 37 e factorial program in Section 3.1.3 illustrates this for a single sequential process. 38 e columns show the contents of Peter’s wallet, the joint account (in Bank1), Paul’s wallet, and Paul’s private account (in Bank2), before and aer each withdrawal (W) and deposit (D). Peter withdraws $10 from Bank1; Paul deposits $5 in Bank2, then withdraws $25 from Bank1. 407 Peter Bank1 Paul Bank2 $7 $100 $5 $300 W D $17 $90 $0 $305 W $17 $65 $25 $305 time Figure 3.30: Concurrent deposits and withdrawals from a joint account in Bank1 and a private account in Bank2. current system produces the same result as if the processes had run sequentially in some order. ere are two important aspects to this re- quirement. First, it does not require the processes to actually run se- quentially, but only to produce results that are the same as if they had run sequentially. For the example in Figure 3.30, the designer of the bank account system can safely allow Paul’s deposit and Peter’s with- drawal to happen concurrently, because the net result will be the same as if the two operations had happened sequentially. Second, there may be more than one possible “correct” result produced by a concurrent program, because we require only that the result be the same as for 408 some sequential order. For example, suppose that Peter and Paul’s joint account starts out with $100, and Peter deposits $40 while Paul concur- rently withdraws half the money in the account. en sequential exe- cution could result in the account balance being either $70 or $90 (see Exercise 3.38).39 ere are still weaker requirements for correct execution of con- current programs. A program for simulating diﬀusion (say, the ﬂow of heat in an object) might consist of a large number of processes, each one representing a small volume of space, that update their values con- currently. Each process repeatedly changes its value to the average of its own value and its neighbors’ values. is algorithm converges to the right answer independent of the order in which the operations are done; there is no need for any restrictions on concurrent use of the shared val- ues. Exercise 3.38: Suppose that Peter, Paul, and Mary share a joint bank account that initially contains $100. Concur- rently, Peter deposits $10, Paul withdraws $20, and Mary withdraws half the money in the account, by executing the following commands: Peter: (set! balance (+ balance 10)) Paul: (set! balance (- balance 20)) Mary: (set! balance (- balance (/ balance 2))) a. List all the diﬀerent possible values for balance aer these three transactions have been completed, assum- 39 A more formal way to express this idea is to say that concurrent programs are inherently nondeterministic. at is, they are described not by single-valued functions, but by functions whose results are sets of possible values. In Section 4.3 we will study a language for expressing nondeterministic computations. 409 ing that the banking system forces the three processes to run sequentially in some order. b. What are some other values that could be produced if the system allows the processes to be interleaved? Draw timing diagrams like the one in Figure 3.29 to explain how these values can occur. 3.4.2 Mechanisms for Controlling Concurrency We’ve seen that the diﬃculty in dealing with concurrent processes is rooted in the need to consider the interleaving of the order of events in the diﬀerent processes. For example, suppose we have two processes, one with three ordered events (a, b, c) and one with three ordered events (x , y, z). If the two processes run concurrently, with no constraints on how their execution is interleaved, then there are 20 diﬀerent possible orderings for the events that are consistent with the individual order- ings for the two processes: (a,b,c,x,y,z) (a,x,b,y,c,z) (x,a,b,c,y,z) (x,a,y,z,b,c) (a,b,x,c,y,z) (a,x,b,y,z,c) (x,a,b,y,c,z) (x,y,a,b,c,z) (a,b,x,y,c,z) (a,x,y,b,c,z) (x,a,b,y,z,c) (x,y,a,b,z,c) (a,b,x,y,z,c) (a,x,y,b,z,c) (x,a,y,b,c,z) (x,y,a,z,b,c) (a,x,b,c,y,z) (a,x,y,z,b,c) (x,a,y,b,z,c) (x,y,z,a,b,c) As programmers designing this system, we would have to consider the eﬀects of each of these 20 orderings and check that each behavior is acceptable. Such an approach rapidly becomes unwieldy as the numbers of processes and events increase. A more practical approach to the design of concurrent systems is to devise general mechanisms that allow us to constrain the interleaving of concurrent processes so that we can be sure that the program behavior 410 is correct. Many mechanisms have been developed for this purpose. In this section, we describe one of them, the serializer. Serializing access to shared state Serialization implements the following idea: Processes will execute con- currently, but there will be certain collections of procedures that cannot be executed concurrently. More precisely, serialization creates distin- guished sets of procedures such that only one execution of a procedure in each serialized set is permied to happen at a time. If some procedure in the set is being executed, then a process that aempts to execute any procedure in the set will be forced to wait until the ﬁrst execution has ﬁnished. We can use serialization to control access to shared variables. For example, if we want to update a shared variable based on the previ- ous value of that variable, we put the access to the previous value of the variable and the assignment of the new value to the variable in the same procedure. We then ensure that no other procedure that assigns to the variable can run concurrently with this procedure by serializing all of these procedures with the same serializer. is guarantees that the value of the variable cannot be changed between an access and the corresponding assignment. Serializers in Scheme To make the above mechanism more concrete, suppose that we have extended Scheme to include a procedure called parallel-execute: (parallel-execute ⟨p1 ⟩ ⟨p2 ⟩ . . . ⟨pk ⟩) Each ⟨p⟩ must be a procedure of no arguments. parallel-execute cre- ates a separate process for each ⟨p⟩, which applies ⟨p⟩ (to no arguments). 411 ese processes all run concurrently.40 As an example of how this is used, consider (define x 10) (parallel-execute (lambda () (set! x (* x x))) (lambda () (set! x (+ x 1)))) is creates two concurrent processes—P1 , which sets x to x times x, and P2 , which increments x. Aer execution is complete, x will be le with one of ﬁve possible values, depending on the interleaving of the events of P1 and P2 : 101: P1 sets x to 100 and then P2 increments x to 101. 121: P2 increments x to 11 and then P1 sets x to x * x. 110: P2 changes x from 10 to 11 between the two times that P1 accesses the value of x during the evaluation of (* x x). 11: P2 accesses x, then P1 sets x to 100, then P2 sets x. 100: P1 accesses x (twice), then P2 sets x to 11, then P1 sets x. We can constrain the concurrency by using serialized procedures, which are created by serializers. Serializers are constructed by make-serializer, whose implementation is given below. A serializer takes a procedure as argument and returns a serialized procedure that behaves like the origi- nal procedure. All calls to a given serializer return serialized procedures in the same set. us, in contrast to the example above, executing (define x 10) 40 parallel-execute is not part of standard Scheme, but it can be implemented in Scheme. In our implementation, the new concurrent processes also run concur- rently with the original Scheme process. Also, in our implementation, the value re- turned by parallel-execute is a special control object that can be used to halt the newly created processes. 412 (define s (make-serializer)) (parallel-execute (s (lambda () (set! x (* x x)))) (s (lambda () (set! x (+ x 1))))) can produce only two possible values for x, 101 or 121. e other pos- sibilities are eliminated, because the execution of P1 and P2 cannot be interleaved. Here is a version of the make-account procedure from Section 3.1.1, where the deposits and withdrawals have been serialized: (define (make-account balance) (define (withdraw amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")) (define (deposit amount) (set! balance (+ balance amount)) balance) (let ((protected (make-serializer))) (define (dispatch m) (cond ((eq? m 'withdraw) (protected withdraw)) ((eq? m 'deposit) (protected deposit)) ((eq? m 'balance) balance) (else (error "Unknown request: MAKE-ACCOUNT" m)))) dispatch)) With this implementation, two processes cannot be withdrawing from or depositing into a single account concurrently. is eliminates the source of the error illustrated in Figure 3.29, where Peter changes the account balance between the times when Paul accesses the balance to compute the new value and when Paul actually performs the assign- 413 ment. On the other hand, each account has its own serializer, so that deposits and withdrawals for diﬀerent accounts can proceed concur- rently. Exercise 3.39: Which of the ﬁve possibilities in the par- allel execution shown above remain if we instead serialize execution as follows: (define x 10) (define s (make-serializer)) (parallel-execute (lambda () (set! x ((s (lambda () (* x x)))))) (s (lambda () (set! x (+ x 1))))) Exercise 3.40: Give all possible values of x that can result from executing (define x 10) (parallel-execute (lambda () (set! x (* x x))) (lambda () (set! x (* x x x)))) Which of these possibilities remain if we instead use seri- alized procedures: (define x 10) (define s (make-serializer)) (parallel-execute (s (lambda () (set! x (* x x)))) (s (lambda () (set! x (* x x x))))) Exercise 3.41: Ben Bitdiddle worries that it would be bet- ter to implement the bank account as follows (where the commented line has been changed): 414 (define (make-account balance) (define (withdraw amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")) (define (deposit amount) (set! balance (+ balance amount)) balance) (let ((protected (make-serializer))) (define (dispatch m) (cond ((eq? m 'withdraw) (protected withdraw)) ((eq? m 'deposit) (protected deposit)) ((eq? m 'balance) ((protected (lambda () balance)))) ; serialized (else (error "Unknown request: MAKE-ACCOUNT" m)))) dispatch)) because allowing unserialized access to the bank balance can result in anomalous behavior. Do you agree? Is there any scenario that demonstrates Ben’s concern? Exercise 3.42: Ben Bitdiddle suggests that it’s a waste of time to create a new serialized procedure in response to every withdraw and deposit message. He says that make- account could be changed so that the calls to protected are done outside the dispatch procedure. at is, an ac- count would return the same serialized procedure (which 415 was created at the same time as the account) each time it is asked for a withdrawal procedure. (define (make-account balance) (define (withdraw amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")) (define (deposit amount) (set! balance (+ balance amount)) balance) (let ((protected (make-serializer))) (let ((protected-withdraw (protected withdraw)) (protected-deposit (protected deposit))) (define (dispatch m) (cond ((eq? m 'withdraw) protected-withdraw) ((eq? m 'deposit) protected-deposit) ((eq? m 'balance) balance) (else (error "Unknown request: MAKE-ACCOUNT" m)))) dispatch))) Is this a safe change to make? In particular, is there any dif- ference in what concurrency is allowed by these two ver- sions of make-account? Complexity of using multiple shared resources Serializers provide a powerful abstraction that helps isolate the com- plexities of concurrent programs so that they can be dealt with carefully and (hopefully) correctly. However, while using serializers is relatively 416 straightforward when there is only a single shared resource (such as a single bank account), concurrent programming can be treacherously diﬃcult when there are multiple shared resources. To illustrate one of the diﬃculties that can arise, suppose we wish to swap the balances in two bank accounts. We access each account to ﬁnd the balance, compute the diﬀerence between the balances, withdraw this diﬀerence from one account, and deposit it in the other account. We could implement this as follows:41 (define (exchange account1 account2) (let ((difference (- (account1 'balance) (account2 'balance)))) ((account1 'withdraw) difference) ((account2 'deposit) difference))) is procedure works well when only a single process is trying to do the exchange. Suppose, however, that Peter and Paul both have access to accounts a1, a2, and a3, and that Peter exchanges a1 and a2 while Paul concurrently exchanges a1 and a3. Even with account deposits and withdrawals serialized for individual accounts (as in the make-account procedure shown above in this section), exchange can still produce in- correct results. For example, Peter might compute the diﬀerence in the balances for a1 and a2, but then Paul might change the balance in a1 before Peter is able to complete the exchange.42 For correct behavior, we must arrange for the exchange procedure to lock out any other con- current accesses to the accounts during the entire time of the exchange. 41 We have simpliﬁed exchange by exploiting the fact that our deposit message ac- cepts negative amounts. (is is a serious bug in our banking system!) 42 If the account balances start out as $10, $20, and $30, then aer any number of concurrent exchanges, the balances should still be $10, $20, and $30 in some order. Serializing the deposits to individual accounts is not suﬃcient to guarantee this. See Exercise 3.43. 417 One way we can accomplish this is by using both accounts’ seri- alizers to serialize the entire exchange procedure. To do this, we will arrange for access to an account’s serializer. Note that we are deliber- ately breaking the modularity of the bank-account object by exposing the serializer. e following version of make-account is identical to the original version given in Section 3.1.1, except that a serializer is pro- vided to protect the balance variable, and the serializer is exported via message passing: (define (make-account-and-serializer balance) (define (withdraw amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")) (define (deposit amount) (set! balance (+ balance amount)) balance) (let ((balance-serializer (make-serializer))) (define (dispatch m) (cond ((eq? m 'withdraw) withdraw) ((eq? m 'deposit) deposit) ((eq? m 'balance) balance) ((eq? m 'serializer) balance-serializer) (else (error "Unknown request: MAKE-ACCOUNT" m)))) dispatch)) We can use this to do serialized deposits and withdrawals. However, unlike our earlier serialized account, it is now the responsibility of each user of bank-account objects to explicitly manage the serialization, for example as follows:43 43 Exercise 3.45 investigates why deposits and withdrawals are no longer automati- cally serialized by the account. 418 (define (deposit account amount) (let ((s (account 'serializer)) (d (account 'deposit))) ((s d) amount))) Exporting the serializer in this way gives us enough ﬂexibility to imple- ment a serialized exchange program. We simply serialize the original exchange procedure with the serializers for both accounts: (define (serialized-exchange account1 account2) (let ((serializer1 (account1 'serializer)) (serializer2 (account2 'serializer))) ((serializer1 (serializer2 exchange)) account1 account2))) Exercise 3.43: Suppose that the balances in three accounts start out as $10, $20, and $30, and that multiple processes run, exchanging the balances in the accounts. Argue that if the processes are run sequentially, aer any number of con- current exchanges, the account balances should be $10, $20, and $30 in some order. Draw a timing diagram like the one in Figure 3.29 to show how this condition can be violated if the exchanges are implemented using the ﬁrst version of the account-exchange program in this section. On the other hand, argue that even with this exchange program, the sum of the balances in the accounts will be preserved. Draw a timing diagram to show how even this condition would be violated if we did not serialize the transactions on individ- ual accounts. Exercise 3.44: Consider the problem of transferring an amount from one account to another. Ben Bitdiddle claims that this 419 can be accomplished with the following procedure, even if there are multiple people concurrently transferring money among multiple accounts, using any account mechanism that serializes deposit and withdrawal transactions, for ex- ample, the version of make-account in the text above. (define (transfer from-account to-account amount) ((from-account 'withdraw) amount) ((to-account 'deposit) amount)) Louis Reasoner claims that there is a problem here, and that we need to use a more sophisticated method, such as the one required for dealing with the exchange problem. Is Louis right? If not, what is the essential diﬀerence between the transfer problem and the exchange problem? (You should assume that the balance in from-account is at least amount.) Exercise 3.45: Louis Reasoner thinks our bank-account sys- tem is unnecessarily complex and error-prone now that de- posits and withdrawals aren’t automatically serialized. He suggests that make-account-and-serializer should have exported the serializer (for use by such procedures as serialized- exchange) in addition to (rather than instead o) using it to serialize accounts and deposits as make-account did. He proposes to redeﬁne accounts as follows: (define (make-account-and-serializer balance) (define (withdraw amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")) (define (deposit amount) (set! balance (+ balance amount)) balance) 420 (let ((balance-serializer (make-serializer))) (define (dispatch m) (cond ((eq? m 'withdraw) (balance-serializer withdraw)) ((eq? m 'deposit) (balance-serializer deposit)) ((eq? m 'balance) balance) ((eq? m 'serializer) balance-serializer) (else (error "Unknown request: MAKE-ACCOUNT" m)))) dispatch)) en deposits are handled as with the original make-account: (define (deposit account amount) ((account 'deposit) amount)) Explain what is wrong with Louis’s reasoning. In particu- lar, consider what happens when serialized-exchange is called. Implementing serializers We implement serializers in terms of a more primitive synchroniza- tion mechanism called a mutex. A mutex is an object that supports two operations—the mutex can be acquired, and the mutex can be released. Once a mutex has been acquired, no other acquire operations on that mutex may proceed until the mutex is released.44 In our implementa- 44 e term “mutex” is an abbreviation for mutual exclusion. e general problem of arranging a mechanism that permits concurrent processes to safely share resources is called the mutual exclusion problem. Our mutex is a simple variant of the semaphore mechanism (see Exercise 3.47), which was introduced in the “THE” Multiprogramming System developed at the Technological University of Eindhoven and named for the university’s initials in Dutch (Dijkstra 1968a). e acquire and release operations were originally called P and V, from the Dutch words passeren (to pass) and vrijgeven (to release), in reference to the semaphores used on railroad systems. Dijkstra’s classic exposition (Dijkstra 1968b) was one of the ﬁrst to clearly present the issues of concur- 421 tion, each serializer has an associated mutex. Given a procedure p, the serializer returns a procedure that acquires the mutex, runs p, and then releases the mutex. is ensures that only one of the procedures pro- duced by the serializer can be running at once, which is precisely the serialization property that we need to guarantee. (define (make-serializer) (let ((mutex (make-mutex))) (lambda (p) (define (serialized-p . args) (mutex 'acquire) (let ((val (apply p args))) (mutex 'release) val)) serialized-p))) e mutex is a mutable object (here we’ll use a one-element list, which we’ll refer to as a cell ) that can hold the value true or false. When the value is false, the mutex is available to be acquired. When the value is true, the mutex is unavailable, and any process that aempts to acquire the mutex must wait. Our mutex constructor make-mutex begins by initializing the cell contents to false. To acquire the mutex, we test the cell. If the mutex is available, we set the cell contents to true and proceed. Otherwise, we wait in a loop, aempting to acquire over and over again, until we ﬁnd that the mutex is available.45 To release the mutex, we set the cell rency control, and showed how to use semaphores to handle a variety of concurrency problems. 45 In most time-shared operating systems, processes that are blocked by a mutex do not waste time “busy-waiting” as above. Instead, the system schedules another process to run while the ﬁrst is waiting, and the blocked process is awakened when the mutex becomes available. 422 contents to false. (define (make-mutex) (let ((cell (list false))) (define (the-mutex m) (cond ((eq? m 'acquire) (if (test-and-set! cell) (the-mutex 'acquire))) ; retry ((eq? m 'release) (clear! cell)))) the-mutex)) (define (clear! cell) (set-car! cell false)) test-and-set! tests the cell and returns the result of the test. In addi- tion, if the test was false, test-and-set! sets the cell contents to true before returning false. We can express this behavior as the following procedure: (define (test-and-set! cell) (if (car cell) true (begin (set-car! cell true) false))) However, this implementation of test-and-set! does not suﬃce as it stands. ere is a crucial subtlety here, which is the essential place where concurrency control enters the system: e test-and-set! op- eration must be performed atomically. at is, we must guarantee that, once a process has tested the cell and found it to be false, the cell con- tents will actually be set to true before any other process can test the cell. If we do not make this guarantee, then the mutex can fail in a way similar to the bank-account failure in Figure 3.29. (See Exercise 3.46.) e actual implementation of test-and-set! depends on the de- tails of how our system runs concurrent processes. For example, we might be executing concurrent processes on a sequential processor us- ing a time-slicing mechanism that cycles through the processes, permit- ting each process to run for a short time before interrupting it and mov- 423 ing on to the next process. In that case, test-and-set! can work by dis- abling time slicing during the testing and seing.46 Alternatively, mul- tiprocessing computers provide instructions that support atomic oper- ations directly in hardware.47 Exercise 3.46: Suppose that we implement test-and-set! using an ordinary procedure as shown in the text, without aempting to make the operation atomic. Draw a timing 46 In Scheme for a single processor, which uses a time-slicing model, test-and- set! can be implemented as follows: (define (test-and-set! cell) (without-interrupts (lambda () (if (car cell) true (begin (set-car! cell true) false))))) without-interrupts disables time-slicing interrupts while its procedure argument is being executed. 47 ere are many variants of such instructions—including test-and-set, test-and- clear, swap, compare-and-exchange, load-reserve, and store-conditional—whose design must be carefully matched to the machine’s processor-memory interface. One issue that arises here is to determine what happens if two processes aempt to acquire the same resource at exactly the same time by using such an instruction. is requires some mechanism for making a decision about which process gets control. Such a mechanism is called an arbiter. Arbiters usually boil down to some sort of hardware device. Un- fortunately, it is possible to prove that one cannot physically construct a fair arbiter that works 100% of the time unless one allows the arbiter an arbitrarily long time to make its decision. e fundamental phenomenon here was originally observed by the fourteenth-century French philosopher Jean Buridan in his commentary on Aristotle’s De caelo. Buridan argued that a perfectly rational dog placed between two equally at- tractive sources of food will starve to death, because it is incapable of deciding which to go to ﬁrst. 424 diagram like the one in Figure 3.29 to demonstrate how the mutex implementation can fail by allowing two processes to acquire the mutex at the same time. Exercise 3.47: A semaphore (of size n) is a generalization of a mutex. Like a mutex, a semaphore supports acquire and release operations, but it is more general in that up to n processes can acquire it concurrently. Additional processes that aempt to acquire the semaphore must wait for release operations. Give implementations of semaphores a. in terms of mutexes b. in terms of atomic test-and-set! operations. Deadlock Now that we have seen how to implement serializers, we can see that account exchanging still has a problem, even with the serialized- exchange procedure above. Imagine that Peter aempts to exchange a1 with a2 while Paul concurrently aempts to exchange a2 with a1. Suppose that Peter’s process reaches the point where it has entered a serialized procedure protecting a1 and, just aer that, Paul’s process en- ters a serialized procedure protecting a2. Now Peter cannot proceed (to enter a serialized procedure protecting a2) until Paul exits the serialized procedure protecting a2. Similarly, Paul cannot proceed until Peter exits the serialized procedure protecting a1. Each process is stalled forever, waiting for the other. is situation is called a deadlock. Deadlock is al- ways a danger in systems that provide concurrent access to multiple shared resources. One way to avoid the deadlock in this situation is to give each ac- count a unique identiﬁcation number and rewrite serialized-exchange 425 so that a process will always aempt to enter a procedure protecting the lowest-numbered account ﬁrst. Although this method works well for the exchange problem, there are other situations that require more so- phisticated deadlock-avoidance techniques, or where deadlock cannot be avoided at all. (See Exercise 3.48 and Exercise 3.49.)48 Exercise 3.48: Explain in detail why the deadlock-avoidance method described above, (i.e., the accounts are numbered, and each process aempts to acquire the smaller-numbered account ﬁrst) avoids deadlock in the exchange problem. Re- write serialized-exchange to incorporate this idea. (You will also need to modify make-account so that each account is created with a number, which can be accessed by sending an appropriate message.) Exercise 3.49: Give a scenario where the deadlock-avoid- ance mechanism described above does not work. (Hint: In the exchange problem, each process knows in advance which accounts it will need to get access to. Consider a situation where a process must get access to some shared resources before it can know which additional shared resources it will require.) 48 e general technique for avoiding deadlock by numbering the shared resources and acquiring them in order is due to Havender (1968). Situations where deadlock can- not be avoided require deadlock-recovery methods, which entail having processes “back out” of the deadlocked state and try again. Deadlock-recovery mechanisms are widely used in database management systems, a topic that is treated in detail in Gray and Reuter 1993. 426 Concurrency, time, and communication We’ve seen how programming concurrent systems requires controlling the ordering of events when diﬀerent processes access shared state, and we’ve seen how to achieve this control through judicious use of serial- izers. But the problems of concurrency lie deeper than this, because, from a fundamental point of view, it’s not always clear what is meant by “shared state.” Mechanisms such as test-and-set! require processes to examine a global shared ﬂag at arbitrary times. is is problematic and ineﬃcient to implement in modern high-speed processors, where due to optimiza- tion techniques such as pipelining and cached memory, the contents of memory may not be in a consistent state at every instant. In con- temporary multiprocessing systems, therefore, the serializer paradigm is being supplanted by new approaches to concurrency control.49 e problematic aspects of shared state also arise in large, distributed systems. For instance, imagine a distributed banking system where indi- vidual branch banks maintain local values for bank balances and period- ically compare these with values maintained by other branches. In such a system the value of “the account balance” would be undetermined, except right aer synchronization. If Peter deposits money in an ac- count he holds jointly with Paul, when should we say that the account balance has changed—when the balance in the local branch changes, or not until aer the synchronization? And if Paul accesses the account 49 One such alternative to serialization is called barrier synchronization. e program- mer permits concurrent processes to execute as they please, but establishes certain synchronization points (“barriers”) through which no process can proceed until all the processes have reached the barrier. Modern processors provide machine instructions that permit programmers to establish synchronization points at places where consis- tency is required. e PowerPC, for example, includes for this purpose two instructions called and (Enforced In-order Execution of Input/Output). 427 from a diﬀerent branch, what are the reasonable constraints to place on the banking system such that the behavior is “correct”? e only thing that might maer for correctness is the behavior observed by Peter and Paul individually and the “state” of the account immediately aer syn- chronization. estions about the “real” account balance or the order of events between synchronizations may be irrelevant or meaningless.50 e basic phenomenon here is that synchronizing diﬀerent pro- cesses, establishing shared state, or imposing an order on events re- quires communication among the processes. In essence, any notion of time in concurrency control must be intimately tied to communica- tion.51 It is intriguing that a similar connection between time and com- munication also arises in the eory of Relativity, where the speed of light (the fastest signal that can be used to synchronize events) is a fundamental constant relating time and space. e complexities we en- counter in dealing with time and state in our computational models may in fact mirror a fundamental complexity of the physical universe. 3.5 Streams We’ve gained a good understanding of assignment as a tool in modeling, as well as an appreciation of the complex problems that assignment raises. It is time to ask whether we could have gone about things in a diﬀerent way, so as to avoid some of these problems. In this section, 50 is may seem like a strange point of view, but there are systems that work this way. International charges to credit-card accounts, for example, are normally cleared on a per-country basis, and the charges made in diﬀerent countries are periodically reconciled. us the account balance may be diﬀerent in diﬀerent countries. 51 For distributed systems, this perspective was pursued by Lamport (1978), who showed how to use communication to establish “global clocks” that can be used to establish orderings on events in distributed systems. 428 we explore an alternative approach to modeling state, based on data structures called streams. As we shall see, streams can mitigate some of the complexity of modeling state. Let’s step back and review where this complexity comes from. In an aempt to model real-world phenomena, we made some apparently reasonable decisions: We modeled real-world objects with local state by computational objects with local variables. We identiﬁed time variation in the real world with time variation in the computer. We implemented the time variation of the states of the model objects in the computer with assignments to the local variables of the model objects. Is there another approach? Can we avoid identifying time in the computer with time in the modeled world? Must we make the model change with time in order to model phenomena in a changing world? ink about the issue in terms of mathematical functions. We can de- scribe the time-varying behavior of a quantity x as a function of time x (t). If we concentrate on x instant by instant, we think of it as a chang- ing quantity. Yet if we concentrate on the entire time history of values, we do not emphasize change—the function itself does not change.52 If time is measured in discrete steps, then we can model a time func- tion as a (possibly inﬁnite) sequence. In this section, we will see how to model change in terms of sequences that represent the time histories of the systems being modeled. To accomplish this, we introduce new data structures called streams. From an abstract point of view, a stream is simply a sequence. However, we will ﬁnd that the straightforward implementation of streams as lists (as in Section 2.2.1) doesn’t fully re- 52 Physicists sometimes adopt this view by introducing the “world lines” of particles as a device for reasoning about motion. We’ve also already mentioned (Section 2.2.3) that this is the natural way to think about signal-processing systems. We will explore applications of streams to signal processing in Section 3.5.3. 429 veal the power of stream processing. As an alternative, we introduce the technique of delayed evaluation, which enables us to represent very large (even inﬁnite) sequences as streams. Stream processing lets us model systems that have state without ever using assignment or mutable data. is has important implications, both theoretical and practical, because we can build models that avoid the drawbacks inherent in introducing assignment. On the other hand, the stream framework raises diﬃculties of its own, and the question of which modeling technique leads to more modular and more easily maintained systems remains open. 3.5.1 Streams Are Delayed Lists As we saw in Section 2.2.3, sequences can serve as standard interfaces for combining program modules. We formulated powerful abstractions for manipulating sequences, such as map, filter, and accumulate, that capture a wide variety of operations in a manner that is both succinct and elegant. Unfortunately, if we represent sequences as lists, this elegance is bought at the price of severe ineﬃciency with respect to both the time and space required by our computations. When we represent manip- ulations on sequences as transformations of lists, our programs must construct and copy data structures (which may be huge) at every step of a process. To see why this is true, let us compare two programs for computing the sum of all the prime numbers in an interval. e ﬁrst program is wrien in standard iterative style:53 53 Assume that we have a predicate prime? (e.g., as in Section 1.2.6) that tests for primality. 430 (define (sum-primes a b) (define (iter count accum) (cond ((> count b) accum) ((prime? count) (iter (+ count 1) (+ count accum))) (else (iter (+ count 1) accum)))) (iter a 0)) e second program performs the same computation using the sequence operations of Section 2.2.3: (define (sum-primes a b) (accumulate + 0 (filter prime? (enumerate-interval a b)))) In carrying out the computation, the ﬁrst program needs to store only the sum being accumulated. In contrast, the ﬁlter in the second pro- gram cannot do any testing until enumerate-interval has constructed a complete list of the numbers in the interval. e ﬁlter generates an- other list, which in turn is passed to accumulate before being collapsed to form a sum. Such large intermediate storage is not needed by the ﬁrst program, which we can think of as enumerating the interval incremen- tally, adding each prime to the sum as it is generated. e ineﬃciency in using lists becomes painfully apparent if we use the sequence paradigm to compute the second prime in the interval from 10,000 to 1,000,000 by evaluating the expression (car (cdr (filter prime? (enumerate-interval 10000 1000000)))) is expression does ﬁnd the second prime, but the computational over- head is outrageous. We construct a list of almost a million integers, ﬁlter 431 this list by testing each element for primality, and then ignore almost all of the result. In a more traditional programming style, we would in- terleave the enumeration and the ﬁltering, and stop when we reached the second prime. Streams are a clever idea that allows one to use sequence manipu- lations without incurring the costs of manipulating sequences as lists. With streams we can achieve the best of both worlds: We can formu- late programs elegantly as sequence manipulations, while aaining the eﬃciency of incremental computation. e basic idea is to arrange to construct a stream only partially, and to pass the partial construction to the program that consumes the stream. If the consumer aempts to access a part of the stream that has not yet been constructed, the stream will automatically construct just enough more of itself to produce the required part, thus preserving the illusion that the entire stream exists. In other words, although we will write programs as if we were process- ing complete sequences, we design our stream implementation to au- tomatically and transparently interleave the construction of the stream with its use. On the surface, streams are just lists with diﬀerent names for the procedures that manipulate them. ere is a constructor, cons-stream, and two selectors, stream-car and stream-cdr, which satisfy the con- straints (stream-car (cons-stream x y)) = x (stream-cdr (cons-stream x y)) = y ere is a distinguishable object, the-empty-stream, which cannot be the result of any cons-stream operation, and which can be identiﬁed with the predicate stream-null?.54 us we can make and use streams, 54 In the implementation, the-empty-stream is the same as the empty list '(), and stream-null? is the same as null?. 432 in just the same way as we can make and use lists, to represent aggregate data arranged in a sequence. In particular, we can build stream analogs of the list operations from Chapter 2, such as list-ref, map, and for- each:55 (define (stream-ref s n) (if (= n 0) (stream-car s) (stream-ref (stream-cdr s) (- n 1)))) (define (stream-map proc s) (if (stream-null? s) the-empty-stream (cons-stream (proc (stream-car s)) (stream-map proc (stream-cdr s))))) (define (stream-for-each proc s) (if (stream-null? s) 'done (begin (proc (stream-car s)) (stream-for-each proc (stream-cdr s))))) stream-for-each is useful for viewing streams: (define (display-stream s) (stream-for-each display-line s)) (define (display-line x) (newline) (display x)) To make the stream implementation automatically and transparently interleave the construction of a stream with its use, we will arrange for 55 is should bother you. e fact that we are deﬁning such similar procedures for streams and lists indicates that we are missing some underlying abstraction. Unfor- tunately, in order to exploit this abstraction, we will need to exert ﬁner control over the process of evaluation than we can at present. We will discuss this point further at the end of Section 3.5.4. In Section 4.2, we’ll develop a framework that uniﬁes lists and streams. 433 the cdr of a stream to be evaluated when it is accessed by the stream- cdr procedure rather than when the stream is constructed by cons- stream. is implementation choice is reminiscent of our discussion of rational numbers in Section 2.1.2, where we saw that we can choose to implement rational numbers so that the reduction of numerator and denominator to lowest terms is performed either at construction time or at selection time. e two rational-number implementations produce the same data abstraction, but the choice has an eﬀect on eﬃciency. ere is a similar relationship between streams and ordinary lists. As a data abstraction, streams are the same as lists. e diﬀerence is the time at which the elements are evaluated. With ordinary lists, both the car and the cdr are evaluated at construction time. With streams, the cdr is evaluated at selection time. Our implementation of streams will be based on a special form called delay. Evaluating (delay ⟨exp⟩) does not evaluate the expression ⟨exp ⟩, but rather returns a so-called delayed object, which we can think of as a “promise” to evaluate ⟨exp ⟩ at some future time. As a companion to delay, there is a procedure called force that takes a delayed object as argument and performs the evaluation—in eﬀect, forcing the delay to fulﬁll its promise. We will see below how delay and force can be im- plemented, but ﬁrst let us use these to construct streams. cons-stream is a special form deﬁned so that (cons-stream ⟨a⟩ ⟨b⟩) is equivalent to (cons ⟨a⟩ (delay ⟨b⟩)) What this means is that we will construct streams using pairs. How- ever, rather than placing the value of the rest of the stream into the cdr of the pair we will put there a promise to compute the rest if it is ever 434 requested. stream-car and stream-cdr can now be deﬁned as proce- dures: (define (stream-car stream) (car stream)) (define (stream-cdr stream) (force (cdr stream))) stream-car selects the car of the pair; stream-cdr selects the cdr of the pair and evaluates the delayed expression found there to obtain the rest of the stream.56 The stream implementation in action To see how this implementation behaves, let us analyze the “outra- geous” prime computation we saw above, reformulated in terms of streams: (stream-car (stream-cdr (stream-filter prime? (stream-enumerate-interval 10000 1000000)))) We will see that it does indeed work eﬃciently. We begin by calling stream-enumerate-interval with the argu- ments 10,000 and 1,000,000. stream-enumerate-interval is the stream analog of enumerate-interval (Section 2.2.3): (define (stream-enumerate-interval low high) (if (> low high) the-empty-stream (cons-stream 56 Although stream-car and stream-cdr can be deﬁned as procedures, cons-stream must be a special form. If cons-stream were a procedure, then, according to our model of evaluation, evaluating (cons-stream ⟨a⟩ ⟨b⟩) would automatically cause ⟨b ⟩ to be evaluated, which is precisely what we do not want to happen. For the same reason, delay must be a special form, though force can be an ordinary procedure. 435 low (stream-enumerate-interval (+ low 1) high)))) and thus the result returned by stream-enumerate-interval, formed by the cons-stream, is57 (cons 10000 (delay (stream-enumerate-interval 10001 1000000))) at is, stream-enumerate-interval returns a stream represented as a pair whose car is 10,000 and whose cdr is a promise to enumerate more of the interval if so requested. is stream is now ﬁltered for primes, using the stream analog of the filter procedure (Section 2.2.3): (define (stream-filter pred stream) (cond ((stream-null? stream) the-empty-stream) ((pred (stream-car stream)) (cons-stream (stream-car stream) (stream-filter pred (stream-cdr stream)))) (else (stream-filter pred (stream-cdr stream))))) stream-filter tests the stream-car of the stream (the car of the pair, which is 10,000). Since this is not prime, stream-filter examines the stream-cdr of its input stream. e call to stream-cdr forces evaluation of the delayed stream-enumerate-interval, which now returns (cons 10001 (delay (stream-enumerate-interval 10002 1000000))) 57 e numbers shown here do not really appear in the delayed expression. What actually appears is the original expression, in an environment in which the variables are bound to the appropriate numbers. For example, (+ low 1) with low bound to 10,000 actually appears where 10001 is shown. 436 stream-filter now looks at the stream-car of this stream, 10,001, sees that this is not prime either, forces another stream-cdr, and so on, until stream-enumerate-interval yields the prime 10,007, whereupon stream-filter, according to its deﬁnition, returns (cons-stream (stream-car stream) (stream-filter pred (stream-cdr stream))) which in this case is (cons 10007 (delay (stream-filter prime? (cons 10008 (delay (stream-enumerate-interval 10009 1000000)))))) is result is now passed to stream-cdr in our original expression. is forces the delayed stream-filter, which in turn keeps forcing the de- layed stream-enumerate-interval until it ﬁnds the next prime, which is 10,009. Finally, the result passed to stream-car in our original ex- pression is (cons 10009 (delay (stream-filter prime? (cons 10010 (delay (stream-enumerate-interval 10011 1000000)))))) stream-car returns 10,009, and the computation is complete. Only as many integers were tested for primality as were necessary to ﬁnd the 437 second prime, and the interval was enumerated only as far as was nec- essary to feed the prime ﬁlter. In general, we can think of delayed evaluation as “demand-driven” programming, whereby each stage in the stream process is activated only enough to satisfy the next stage. What we have done is to decouple the actual order of events in the computation from the apparent struc- ture of our procedures. We write procedures as if the streams existed “all at once” when, in reality, the computation is performed incrementally, as in traditional programming styles. Implementing delay and force Although delay and force may seem like mysterious operations, their implementation is really quite straightforward. delay must package an expression so that it can be evaluated later on demand, and we can ac- complish this simply by treating the expression as the body of a proce- dure. delay can be a special form such that (delay ⟨exp⟩) is syntactic sugar for (lambda () ⟨exp⟩) force simply calls the procedure (of no arguments) produced by delay, so we can implement force as a procedure: (define (force delayed-object) (delayed-object)) is implementation suﬃces for delay and force to work as advertised, but there is an important optimization that we can include. In many ap- plications, we end up forcing the same delayed object many times. is can lead to serious ineﬃciency in recursive programs involving streams. (See Exercise 3.57.) e solution is to build delayed objects so that the 438 ﬁrst time they are forced, they store the value that is computed. Subse- quent forcings will simply return the stored value without repeating the computation. In other words, we implement delay as a special-purpose memoized procedure similar to the one described in Exercise 3.27. One way to accomplish this is to use the following procedure, which takes as argument a procedure (of no arguments) and returns a memoized ver- sion of the procedure. e ﬁrst time the memoized procedure is run, it saves the computed result. On subsequent evaluations, it simply returns the result. (define (memo-proc proc) (let ((already-run? false) (result false)) (lambda () (if (not already-run?) (begin (set! result (proc)) (set! already-run? true) result) result)))) delay is then deﬁned so that (delay ⟨exp⟩) is equivalent to (memo-proc (lambda () ⟨exp⟩)) and force is as deﬁned previously.58 58 ere are many possible implementations of streams other than the one described in this section. Delayed evaluation, which is the key to making streams practical, was inherent in Algol 60’s call-by-name parameter-passing method. e use of this mech- anism to implement streams was ﬁrst described by Landin (1965). Delayed evaluation for streams was introduced into Lisp by Friedman and Wise (1976). In their implemen- tation, cons always delays evaluating its arguments, so that lists automatically behave as streams. e memoizing optimization is also known as call-by-need. e Algol com- munity would refer to our original delayed objects as call-by-name thunks and to the optimized versions as call-by-need thunks. 439 Exercise 3.50: Complete the following deﬁnition, which generalizes stream-map to allow procedures that take mul- tiple arguments, analogous to map in Section 2.2.1, Footnote 12. (define (stream-map proc . argstreams) (if (⟨??⟩ (car argstreams)) the-empty-stream (⟨??⟩ (apply proc (map ⟨??⟩ argstreams)) (apply stream-map (cons proc (map ⟨??⟩ argstreams)))))) Exercise 3.51: In order to take a closer look at delayed eval- uation, we will use the following procedure, which simply returns its argument aer printing it: (define (show x) (display-line x) x) What does the interpreter print in response to evaluating each expression in the following sequence?59 (define x 59 Exercises such as Exercise 3.51 and Exercise 3.52 are valuable for testing our un- derstanding of how delay works. On the other hand, intermixing delayed evaluation with printing—and, even worse, with assignment—is extremely confusing, and instruc- tors of courses on computer languages have traditionally tormented their students with examination questions such as the ones in this section. Needless to say, writing pro- grams that depend on such subtleties is odious programming style. Part of the power of stream processing is that it lets us ignore the order in which events actually happen in our programs. Unfortunately, this is precisely what we cannot aﬀord to do in the presence of assignment, which forces us to be concerned with time and change. 440 (stream-map show (stream-enumerate-interval 0 10))) (stream-ref x 5) (stream-ref x 7) Exercise 3.52: Consider the sequence of expressions (define sum 0) (define (accum x) (set! sum (+ x sum)) sum) (define seq (stream-map accum (stream-enumerate-interval 1 20))) (define y (stream-filter even? seq)) (define z (stream-filter (lambda (x) (= (remainder x 5) 0)) seq)) (stream-ref y 7) (display-stream z) What is the value of sum aer each of the above expressions is evaluated? What is the printed response to evaluating the stream-ref and display-stream expressions? Would these responses diﬀer if we had implemented (delay ⟨exp⟩) simply as (lambda () ⟨exp⟩) without using the optimiza- tion provided by memo-proc? Explain. 3.5.2 Infinite Streams We have seen how to support the illusion of manipulating streams as complete entities even though, in actuality, we compute only as much of the stream as we need to access. We can exploit this technique to rep- resent sequences eﬃciently as streams, even if the sequences are very 441 long. What is more striking, we can use streams to represent sequences that are inﬁnitely long. For instance, consider the following deﬁnition of the stream of positive integers: (define (integers-starting-from n) (cons-stream n (integers-starting-from (+ n 1)))) (define integers (integers-starting-from 1)) is makes sense because integers will be a pair whose car is 1 and whose cdr is a promise to produce the integers beginning with 2. is is an inﬁnitely long stream, but in any given time we can examine only a ﬁnite portion of it. us, our programs will never know that the entire inﬁnite stream is not there. Using integers we can deﬁne other inﬁnite streams, such as the stream of integers that are not divisible by 7: (define (divisible? x y) (= (remainder x y) 0)) (define no-sevens (stream-filter (lambda (x) (not (divisible? x 7))) integers)) en we can ﬁnd integers not divisible by 7 simply by accessing ele- ments of this stream: (stream-ref no-sevens 100) 117 In analogy with integers, we can deﬁne the inﬁnite stream of Fibonacci numbers: (define (fibgen a b) (cons-stream a (fibgen b (+ a b)))) (define fibs (fibgen 0 1)) fibs is a pair whose car is 0 and whose cdr is a promise to evaluate (fibgen 1 1). When we evaluate this delayed (fibgen 1 1), it will 442 produce a pair whose car is 1 and whose cdr is a promise to evaluate (fibgen 1 2), and so on. For a look at a more exciting inﬁnite stream, we can generalize the no-sevens example to construct the inﬁnite stream of prime numbers, using a method known as the sieve of Eratosthenes.60 We start with the integers beginning with 2, which is the ﬁrst prime. To get the rest of the primes, we start by ﬁltering the multiples of 2 from the rest of the integers. is leaves a stream beginning with 3, which is the next prime. Now we ﬁlter the multiples of 3 from the rest of this stream. is leaves a stream beginning with 5, which is the next prime, and so on. In other words, we construct the primes by a sieving process, described as fol- lows: To sieve a stream S, form a stream whose ﬁrst element is the ﬁrst element of S and the rest of which is obtained by ﬁltering all multiples of the ﬁrst element of S out of the rest of S and sieving the result. is process is readily described in terms of stream operations: (define (sieve stream) (cons-stream (stream-car stream) (sieve (stream-filter (lambda (x) (not (divisible? x (stream-car stream)))) (stream-cdr stream))))) (define primes (sieve (integers-starting-from 2))) 60 Eratosthenes, a third-century .. Alexandrian Greek philosopher, is famous for giving the ﬁrst accurate estimate of the circumference of the Earth, which he computed by observing shadows cast at noon on the day of the summer solstice. Eratosthenes’s sieve method, although ancient, has formed the basis for special-purpose hardware “sieves” that, until recently, were the most powerful tools in existence for locating large primes. Since the 70s, however, these methods have been superseded by outgrowths of the probabilistic techniques discussed in Section 1.2.6. 443 sieve car cons cdr filter: sieve not divisible? Figure 3.31: e prime sieve viewed as a signal-processing system. Now to ﬁnd a particular prime we need only ask for it: (stream-ref primes 50) 233 It is interesting to contemplate the signal-processing system set up by sieve, shown in the “Henderson diagram” in Figure 3.31.61 e input stream feeds into an “unconser” that separates the ﬁrst element of the stream from the rest of the stream. e ﬁrst element is used to construct a divisibility ﬁlter, through which the rest is passed, and the output of the ﬁlter is fed to another sieve box. en the original ﬁrst element is consed onto the output of the internal sieve to form the output stream. us, not only is the stream inﬁnite, but the signal processor is also inﬁnite, because the sieve contains a sieve within it. 61 We have named these ﬁgures aer Peter Henderson, who was the ﬁrst person to show us diagrams of this sort as a way of thinking about stream processing. Each solid line represents a stream of values being transmied. e dashed line from the car to the cons and the filter indicates that this is a single value rather than a stream. 444 Defining streams implicitly e integers and fibs streams above were deﬁned by specifying “gen- erating” procedures that explicitly compute the stream elements one by one. An alternative way to specify streams is to take advantage of de- layed evaluation to deﬁne streams implicitly. For example, the following expression deﬁnes the stream ones to be an inﬁnite stream of ones: (define ones (cons-stream 1 ones)) is works much like the deﬁnition of a recursive procedure: ones is a pair whose car is 1 and whose cdr is a promise to evaluate ones. Evaluating the cdr gives us again a 1 and a promise to evaluate ones, and so on. We can do more interesting things by manipulating streams with operations such as add-streams, which produces the elementwise sum of two given streams:62 (define (add-streams s1 s2) (stream-map + s1 s2)) Now we can deﬁne the integers as follows: (define integers (cons-stream 1 (add-streams ones integers))) is deﬁnes integers to be a stream whose ﬁrst element is 1 and the rest of which is the sum of ones and integers. us, the second element of integers is 1 plus the ﬁrst element of integers, or 2; the third element of integers is 1 plus the second element of integers, or 3; and so on. is deﬁnition works because, at any point, enough of the integers stream has been generated so that we can feed it back into the deﬁnition to produce the next integer. We can deﬁne the Fibonacci numbers in the same style: 62 is uses the generalized version of stream-map from Exercise 3.50. 445 (define fibs (cons-stream 0 (cons-stream 1 (add-streams (stream-cdr fibs) fibs)))) is deﬁnition says that fibs is a stream beginning with 0 and 1, such that the rest of the stream can be generated by adding fibs to itself shied by one place: 1 1 2 3 . . . = (stream-cdr fibs) 5 8 13 21 0 1 1 2 . . . = fibs 3 5 8 13 0 1 1 2 3 5 8 13 21 34 . . . = fibs scale-stream is another useful procedure in formulating such stream deﬁnitions. is multiplies each item in a stream by a given constant: (define (scale-stream stream factor) (stream-map (lambda (x) (* x factor)) stream)) For example, (define double (cons-stream 1 (scale-stream double 2))) produces the stream of powers of 2: 1, 2, 4, 8, 16, 32, . . .. An alternate deﬁnition of the stream of primes can be given by start- ing with the integers and ﬁltering them by testing for primality. We will need the ﬁrst prime, 2, to get started: (define primes (cons-stream 2 (stream-filter prime? (integers-starting-from 3)))) is deﬁnition is not so straightforward as it appears, because we will test whether a number n is prime by checking whether n is divisible by √ a prime (not by just any integer) less than or equal to n: 446 (define (prime? n) (define (iter ps) (cond ((> (square (stream-car ps)) n) true) ((divisible? n (stream-car ps)) false) (else (iter (stream-cdr ps))))) (iter primes)) is is a recursive deﬁnition, since primes is deﬁned in terms of the prime? predicate, which itself uses the primes stream. e reason this procedure works is that, at any point, enough of the primes stream has been generated to test the primality of the numbers we need to check next. at is, for every n we test for primality, either n is not prime (in which case there is a prime already generated that divides it) or n is prime (in which case there is a prime already generated—i.e., a prime √ less than n—that is greater than n).63 Exercise 3.53: Without running the program, describe the elements of the stream deﬁned by (define s (cons-stream 1 (add-streams s s))) Exercise 3.54: Deﬁne a procedure mul-streams, analogous to add-streams, that produces the elementwise product of its two input streams. Use this together with the stream of integers to complete the following deﬁnition of the stream whose n th element (counting from 0) is n + 1 factorial: 63 is last point is very subtle and relies on the fact that p ≤ pn2 . (Here, p k denotes n+1 the k thprime.) Estimates such as these are very diﬃcult to establish. e ancient proof by Euclid that there are an inﬁnite number of primes shows that pn+1 ≤ p 1 p 2 · · · pn + 1, and no substantially beer result was proved until 1851, when the Russian mathemati- cian P. L. Chebyshev established that pn+1 ≤ 2pn for all n. is result, originally con- jectured in 1845, is known as Bertrand’s hypothesis. A proof can be found in section 22.3 of Hardy and Wright 1960. 447 (define factorials (cons-stream 1 (mul-streams ⟨??⟩ ⟨??⟩))) Exercise 3.55: Deﬁne a procedure partial-sums that takes as argument a stream S and returns the stream whose ele- ments are S 0 , S 0 +S 1 , S 0 +S 1 +S 2 , . . .. For example, (partial- sums integers) should be the stream 1, 3, 6, 10, 15, . . .. Exercise 3.56: A famous problem, ﬁrst raised by R. Ham- ming, is to enumerate, in ascending order with no repeti- tions, all positive integers with no prime factors other than 2, 3, or 5. One obvious way to do this is to simply test each integer in turn to see whether it has any factors other than 2, 3, and 5. But this is very ineﬃcient, since, as the integers get larger, fewer and fewer of them ﬁt the requirement. As an alternative, let us call the required stream of numbers S and notice the following facts about it. • S begins with 1. • e elements of (scale-stream S 2) are also ele- ments of S. • e same is true for (scale-stream S 3) and (scale- stream 5 S). • ese are all the elements of S. Now all we have to do is combine elements from these sources. For this we deﬁne a procedure merge that combines two or- dered streams into one ordered result stream, eliminating repetitions: 448 (define (merge s1 s2) (cond ((stream-null? s1) s2) ((stream-null? s2) s1) (else (let ((s1car (stream-car s1)) (s2car (stream-car s2))) (cond ((< s1car s2car) (cons-stream s1car (merge (stream-cdr s1) s2))) ((> s1car s2car) (cons-stream s2car (merge s1 (stream-cdr s2)))) (else (cons-stream s1car (merge (stream-cdr s1) (stream-cdr s2))))))))) en the required stream may be constructed with merge, as follows: (define S (cons-stream 1 (merge ⟨??⟩ ⟨??⟩))) Fill in the missing expressions in the places marked ⟨⁇⟩ above. Exercise 3.57: How many additions are performed when we compute the n th Fibonacci number using the deﬁnition of fibs based on the add-streams procedure? Show that the number of additions would be exponentially greater if we had implemented (delay ⟨exp⟩) simply as (lambda 449 () ⟨exp⟩), without using the optimization provided by the memo-proc procedure described in Section 3.5.1.64 Exercise 3.58: Give an interpretation of the stream com- puted by the following procedure: (define (expand num den radix) (cons-stream (quotient (* num radix) den) (expand (remainder (* num radix) den) den radix))) (quotient is a primitive that returns the integer quotient of two integers.) What are the successive elements produced by (expand 1 7 10)? What is produced by (expand 3 8 10)? Exercise 3.59: In Section 2.5.3 we saw how to implement a polynomial arithmetic system representing polynomials as lists of terms. In a similar way, we can work with power series, such as x2 x3 x4 e = 1+x + x + + +..., 2 3·2 4·3·2 x2 x4 cos x = 1 − + −..., 2 4·3·2 x3 x5 sin x = x − + −... 3·2 5·4·3·2 64 is exercise shows how call-by-need is closely related to ordinary memoization as described in Exercise 3.27. In that exercise, we used assignment to explicitly construct a local table. Our call-by-need stream optimization eﬀectively constructs such a table automatically, storing values in the previously forced parts of the stream. 450 represented as inﬁnite streams. We will represent the series a 0 + a 1x + a 2x 2 + a 3x 3 + . . . as the stream whose elements are the coeﬃcients a 0 , a 1 , a 2 , a 3 , . . .. a. e integral of the series a 0 + a 1x + a 2x 2 + a 3x 3 + . . . is the series 1 1 1 c + a 0x + a 1x 2 + a 2x 3 + a 3x 4 + . . . , 2 3 4 where c is any constant. Deﬁne a procedure integrate- series that takes as input a stream a 0 , a 1 , a 2 , . . . rep- resenting a power series and returns the stream a 0 , 1 a , 1 a , . . . of coeﬃcients of the non-constant terms 2 1 3 2 of the integral of the series. (Since the result has no constant term, it doesn’t represent a power series; when we use integrate-series, we will cons on the ap- propriate constant.) b. e function x 7→ e x is its own derivative. is im- plies that e x and the integral of e x are the same se- ries, except for the constant term, which is e 0 = 1. Accordingly, we can generate the series for e x as (define exp-series (cons-stream 1 (integrate-series exp-series))) Show how to generate the series for sine and cosine, starting from the facts that the derivative of sine is cosine and the derivative of cosine is the negative of sine: (define cosine-series (cons-stream 1 ⟨??⟩)) (define sine-series (cons-stream 0 ⟨??⟩)) 451 Exercise 3.60: With power series represented as streams of coeﬃcients as in Exercise 3.59, adding series is imple- mented by add-streams. Complete the deﬁnition of the fol- lowing procedure for multiplying series: (define (mul-series s1 s2) (cons-stream ⟨??⟩ (add-streams ⟨??⟩ ⟨??⟩))) You can test your procedure by verifying that sin2 x + cos2 x = 1, using the series from Exercise 3.59. Exercise 3.61: Let S be a power series (Exercise 3.59) whose constant term is 1. Suppose we want to ﬁnd the power se- ries 1/S, that is, the series X such that S X = 1. Write S = 1 + S R where S R is the part of S aer the constant term. en we can solve for X as follows: S·X = 1, (1 + S R ) · X = 1, X + SR · X = 1, X = 1 − SR · X . In other words, X is the power series whose constant term is 1 and whose higher-order terms are given by the negative of S R times X . Use this idea to write a procedure invert- unit-series that computes 1/S for a power series S with constant term 1. You will need to use mul-series from Ex- ercise 3.60. Exercise 3.62: Use the results of Exercise 3.60 and Exer- cise 3.61 to deﬁne a procedure div-series that divides two power series. div-series should work for any two series, 452 provided that the denominator series begins with a nonzero constant term. (If the denominator has a zero constant term, then div-series should signal an error.) Show how to use div-series together with the result of Exercise 3.59 to gen- erate the power series for tangent. 3.5.3 Exploiting the Stream Paradigm Streams with delayed evaluation can be a powerful modeling tool, pro- viding many of the beneﬁts of local state and assignment. Moreover, they avoid some of the theoretical tangles that accompany the intro- duction of assignment into a programming language. e stream approach can be illuminating because it allows us to build systems with diﬀerent module boundaries than systems organized around assignment to state variables. For example, we can think of an entire time series (or signal) as a focus of interest, rather than the values of the state variables at individual moments. is makes it convenient to combine and compare components of state from diﬀerent moments. Formulating iterations as stream processes In Section 1.2.1, we introduced iterative processes, which proceed by updating state variables. We know now that we can represent state as a “timeless” stream of values rather than as a set of variables to be up- dated. Let’s adopt this perspective in revisiting the square-root proce- dure from Section 1.1.7. Recall that the idea is to generate a sequence of beer and beer guesses for the square root of x by applying over and over again the procedure that improves guesses: (define (sqrt-improve guess x) (average guess (/ x guess))) 453 In our original sqrt procedure, we made these guesses be the successive values of a state variable. Instead we can generate the inﬁnite stream of guesses, starting with an initial guess of 1:65 (define (sqrt-stream x) (define guesses (cons-stream 1.0 (stream-map (lambda (guess) (sqrt-improve guess x)) guesses))) guesses) (display-stream (sqrt-stream 2)) 1. 1.5 1.4166666666666665 1.4142156862745097 1.4142135623746899 ... We can generate more and more terms of the stream to get beer and beer guesses. If we like, we can write a procedure that keeps generating terms until the answer is good enough. (See Exercise 3.64.) Another iteration that we can treat in the same way is to generate an approximation to π , based upon the alternating series that we saw in Section 1.3.1: π 1 1 1 = 1− + − +.... 4 3 5 7 We ﬁrst generate the stream of summands of the series (the reciprocals of the odd integers, with alternating signs). en we take the stream of 65 We can’t use let to bind the local variable guesses, because the value of guesses depends on guesses itself. Exercise 3.63 addresses why we want a local variable here. 454 sums of more and more terms (using the partial-sums procedure of Exercise 3.55) and scale the result by 4: (define (pi-summands n) (cons-stream (/ 1.0 n) (stream-map - (pi-summands (+ n 2))))) (define pi-stream (scale-stream (partial-sums (pi-summands 1)) 4)) (display-stream pi-stream) 4. 2.666666666666667 3.466666666666667 2.8952380952380956 3.3396825396825403 2.9760461760461765 3.2837384837384844 3.017071817071818 ... is gives us a stream of beer and beer approximations to π , although the approximations converge rather slowly. Eight terms of the sequence bound the value of π between 3.284 and 3.017. So far, our use of the stream of states approach is not much diﬀerent from updating state variables. But streams give us an opportunity to do some interesting tricks. For example, we can transform a stream with a sequence accelerator that converts a sequence of approximations to a new sequence that converges to the same value as the original, only faster. One such accelerator, due to the eighteenth-century Swiss math- ematician Leonhard Euler, works well with sequences that are partial sums of alternating series (series of terms with alternating signs). In Eu- ler’s technique, if Sn is the n th term of the original sum sequence, then 455 the accelerated sequence has terms (Sn+1 − Sn )2 Sn+1 − . Sn −1 − 2Sn + Sn+1 us, if the original sequence is represented as a stream of values, the transformed sequence is given by (define (euler-transform s) (let ((s0 (stream-ref s 0)) ; Sn −1 (s1 (stream-ref s 1)) ; Sn (s2 (stream-ref s 2))) ; Sn+1 (cons-stream (- s2 (/ (square (- s2 s1)) (+ s0 (* -2 s1) s2))) (euler-transform (stream-cdr s))))) We can demonstrate Euler acceleration with our sequence of approxi- mations to π : (display-stream (euler-transform pi-stream)) 3.166666666666667 3.1333333333333337 3.1452380952380956 3.13968253968254 3.1427128427128435 3.1408813408813416 3.142071817071818 3.1412548236077655 ... Even beer, we can accelerate the accelerated sequence, and recursively accelerate that, and so on. Namely, we create a stream of streams (a structure we’ll call a tableau) in which each stream is the transform of the preceding one: 456 (define (make-tableau transform s) (cons-stream s (make-tableau transform (transform s)))) e tableau has the form s 00 s 01 s 02 s 03 s 04 . . . s 10 s 11 s 12 s 13 . . . s 20 s 21 s 22 . . . ... Finally, we form a sequence by taking the ﬁrst term in each row of the tableau: (define (accelerated-sequence transform s) (stream-map stream-car (make-tableau transform s))) We can demonstrate this kind of “super-acceleration” of the π sequence: (display-stream (accelerated-sequence euler-transform pi-stream)) 4. 3.166666666666667 3.142105263157895 3.141599357319005 3.1415927140337785 3.1415926539752927 3.1415926535911765 3.141592653589778 ... e result is impressive. Taking eight terms of the sequence yields the correct value of π to 14 decimal places. If we had used only the original π sequence, we would need to compute on the order of 1013 terms (i.e., expanding the series far enough so that the individual terms are less than 10 −13 ) to get that much accuracy! 457 We could have implemented these acceleration techniques without using streams. But the stream formulation is particularly elegant and convenient because the entire sequence of states is available to us as a data structure that can be manipulated with a uniform set of operations. Exercise 3.63: Louis Reasoner asks why the sqrt-stream procedure was not wrien in the following more straight- forward way, without the local variable guesses: (define (sqrt-stream x) (cons-stream 1.0 (stream-map (lambda (guess) (sqrt-improve guess x)) (sqrt-stream x)))) Alyssa P. Hacker replies that this version of the procedure is considerably less eﬃcient because it performs redundant computation. Explain Alyssa’s answer. Would the two ver- sions still diﬀer in eﬃciency if our implementation of delay used only (lambda () ⟨exp⟩) without using the optimiza- tion provided by memo-proc (Section 3.5.1)? Exercise 3.64: Write a procedure stream-limit that takes as arguments a stream and a number (the tolerance). It should examine the stream until it ﬁnds two successive elements that diﬀer in absolute value by less than the tolerance, and return the second of the two elements. Using this, we could compute square roots up to a given tolerance by (define (sqrt x tolerance) (stream-limit (sqrt-stream x) tolerance)) 458 Exercise 3.65: Use the series 1 1 1 ln 2 = 1 − + − + . . . 2 3 4 to compute three sequences of approximations to the nat- ural logarithm of 2, in the same way we did above for π . How rapidly do these sequences converge? Infinite streams of pairs In Section 2.2.3, we saw how the sequence paradigm handles traditional nested loops as processes deﬁned on sequences of pairs. If we generalize this technique to inﬁnite streams, then we can write programs that are not easily represented as loops, because the “looping” must range over an inﬁnite set. For example, suppose we want to generalize the prime-sum-pairs procedure of Section 2.2.3 to produce the stream of pairs of all integers (i, j) with i ≤ j such that i + j is prime. If int-pairs is the sequence of all pairs of integers (i, j) with i ≤ j, then our required stream is simply66 (stream-filter (lambda (pair) (prime? (+ (car pair) (cadr pair)))) int-pairs) Our problem, then, is to produce the stream int-pairs. More generally, suppose we have two streams S = (Si ) and T = (T j ), and imagine the inﬁnite rectangular array (S 0 , T0 ) (S 0 , T1 ) (S 0 , T2 ) . . . (S 1 , T0 ) (S 1 , T1 ) (S 1 , T2 ) . . . (S 2 , T0 ) (S 2 , T1 ) (S 2 , T2 ) . . . ... 66 As in Section 2.2.3, we represent a pair of integers as a list rather than a Lisp pair. 459 We wish to generate a stream that contains all the pairs in the array that lie on or above the diagonal, i.e., the pairs (S 0 , T0 ) (S 0 , T1 ) (S 0 , T2 ) . . . (S 1 , T1 ) (S 1 , T2 ) . . . (S 2 , T2 ) . . . ... (If we take both S and T to be the stream of integers, then this will be our desired stream int-pairs.) Call the general stream of pairs (pairs S T), and consider it to be composed of three parts: the pair (S 0 , T0 ), the rest of the pairs in the ﬁrst row, and the remaining pairs:67 (S 0 , T0 ) (S 0 , T1 ) (S 0 , T2 ) . . . (S 1 , T1 ) (S 1 , T2 ) . . . (S 2 , T2 ) . . . ... Observe that the third piece in this decomposition (pairs that are not in the ﬁrst row) is (recursively) the pairs formed from (stream-cdr S) and (stream-cdr T). Also note that the second piece (the rest of the ﬁrst row) is (stream-map (lambda (x) (list (stream-car s) x)) (stream-cdr t)) us we can form our stream of pairs as follows: (define (pairs s t) (cons-stream (list (stream-car s) (stream-car t)) 67 See Exercise 3.68 for some insight into why we chose this decomposition. 460 (⟨combine-in-some-way⟩ (stream-map (lambda (x) (list (stream-car s) x)) (stream-cdr t)) (pairs (stream-cdr s) (stream-cdr t))))) In order to complete the procedure, we must choose some way to com- bine the two inner streams. One idea is to use the stream analog of the append procedure from Section 2.2.1: (define (stream-append s1 s2) (if (stream-null? s1) s2 (cons-stream (stream-car s1) (stream-append (stream-cdr s1) s2)))) is is unsuitable for inﬁnite streams, however, because it takes all the elements from the ﬁrst stream before incorporating the second stream. In particular, if we try to generate all pairs of positive integers using (pairs integers integers) our stream of results will ﬁrst try to run through all pairs with the ﬁrst integer equal to 1, and hence will never produce pairs with any other value of the ﬁrst integer. To handle inﬁnite streams, we need to devise an order of combina- tion that ensures that every element will eventually be reached if we let our program run long enough. An elegant way to accomplish this is with the following interleave procedure:68 68 e precise statement of the required property on the order of combination is as follows: ere should be a function f of two arguments such that the pair correspond- ing to element i of the ﬁrst stream and element j of the second stream will appear as element number f (i, j) of the output stream. e trick of using interleave to accom- plish this was shown to us by David Turner, who employed it in the language KRC (Turner 1981). 461 (define (interleave s1 s2) (if (stream-null? s1) s2 (cons-stream (stream-car s1) (interleave s2 (stream-cdr s1))))) Since interleave takes elements alternately from the two streams, ev- ery element of the second stream will eventually ﬁnd its way into the interleaved stream, even if the ﬁrst stream is inﬁnite. We can thus generate the required stream of pairs as (define (pairs s t) (cons-stream (list (stream-car s) (stream-car t)) (interleave (stream-map (lambda (x) (list (stream-car s) x)) (stream-cdr t)) (pairs (stream-cdr s) (stream-cdr t))))) Exercise 3.66: Examine the stream (pairs integers integers). Can you make any general comments about the order in which the pairs are placed into the stream? For example, approximately how many pairs precede the pair (1, 100)? the pair (99, 100)? the pair (100, 100)? (If you can make pre- cise mathematical statements here, all the beer. But feel free to give more qualitative answers if you ﬁnd yourself geing bogged down.) Exercise 3.67: Modify the pairs procedure so that (pairs integers integers) will produce the stream of all pairs of integers (i, j) (without the condition i ≤ j). Hint: You will need to mix in an additional stream. 462 Exercise 3.68: Louis Reasoner thinks that building a stream of pairs from three parts is unnecessarily complicated. In- stead of separating the pair (S 0 , T0 ) from the rest of the pairs in the ﬁrst row, he proposes to work with the whole ﬁrst row, as follows: (define (pairs s t) (interleave (stream-map (lambda (x) (list (stream-car s) x)) t) (pairs (stream-cdr s) (stream-cdr t)))) Does this work? Consider what happens if we evaluate (pairs integers integers) using Louis’s deﬁnition of pairs. Exercise 3.69: Write a procedure triples that takes three inﬁnite streams, S, T , and U , and produces the stream of triples (Si , T j , Uk ) such that i ≤ j ≤ k. Use triples to gen- erate the stream of all Pythagorean triples of positive inte- gers, i.e., the triples (i, j, k) such that i ≤ j and i 2 + j 2 = k 2 . Exercise 3.70: It would be nice to be able to generate streams in which the pairs appear in some useful order, rather than in the order that results from an ad hoc interleaving pro- cess. We can use a technique similar to the merge procedure of Exercise 3.56, if we deﬁne a way to say that one pair of integers is “less than” another. One way to do this is to de- ﬁne a “weighting function” W (i, j) and stipulate that (i 1 , j 1 ) is less than (i 2 , j 2 ) if W (i 1 , j 1 ) < W (i 2 , j 2 ). Write a proce- dure merge-weighted that is like merge, except that merge- weighted takes an additional argument weight, which is a procedure that computes the weight of a pair, and is used 463 to determine the order in which elements should appear in the resulting merged stream.69 Using this, generalize pairs to a procedure weighted-pairs that takes two streams, to- gether with a procedure that computes a weighting func- tion, and generates the stream of pairs, ordered according to weight. Use your procedure to generate a. the stream of all pairs of positive integers (i, j) with i ≤ j ordered according to the sum i + j, b. the stream of all pairs of positive integers (i, j) with i ≤ j, where neither i nor j is divisible by 2, 3, or 5, and the pairs are ordered according to the sum 2i + 3j + 5ij. Exercise 3.71: Numbers that can be expressed as the sum of two cubes in more than one way are sometimes called Ra- manujan numbers, in honor of the mathematician Srinivasa Ramanujan.70 Ordered streams of pairs provide an elegant solution to the problem of computing these numbers. To ﬁnd a number that can be wrien as the sum of two cubes in two diﬀerent ways, we need only generate the stream of pairs of integers (i, j) weighted according to the sum i 3 + j 3 69 We will require that the weighting function be such that the weight of a pair in- creases as we move out along a row or down along a column of the array of pairs. 70 To quote from G. H. Hardy’s obituary of Ramanujan (Hardy 1921): “It was Mr. Lilewood (I believe) who remarked that ‘every positive integer was one of his friends.’ I remember once going to see him when he was lying ill at Putney. I had ridden in taxi- cab No. 1729, and remarked that the number seemed to me a rather dull one, and that I hoped it was not an unfavorable omen. ‘No,’ he replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two diﬀerent ways.’ ” e trick of using weighted pairs to generate the Ramanujan numbers was shown to us by Charles Leiserson. 464 (see Exercise 3.70), then search the stream for two consecu- tive pairs with the same weight. Write a procedure to gener- ate the Ramanujan numbers. e ﬁrst such number is 1,729. What are the next ﬁve? Exercise 3.72: In a similar way to Exercise 3.71 generate a stream of all numbers that can be wrien as the sum of two squares in three diﬀerent ways (showing how they can be so wrien). Streams as signals We began our discussion of streams by describing them as computa- tional analogs of the “signals” in signal-processing systems. In fact, we can use streams to model signal-processing systems in a very direct way, representing the values of a signal at successive time intervals as consecutive elements of a stream. For instance, we can implement an integrator or summer that, for an input stream x = (xi ), an initial value C, and a small increment dt, accumulates the sum ∑ i Si = C + x j dt j =1 and returns the stream of values S = (Si ). e following integral pro- cedure is reminiscent of the “implicit style” deﬁnition of the stream of integers (Section 3.5.2): (define (integral integrand initial-value dt) (define int (cons-stream initial-value (add-streams (scale-stream integrand dt) int))) int) 465 initial-value input integral scale: dt cons add Figure 3.32: e integral procedure viewed as a signal- processing system. Figure 3.32 is a picture of a signal-processing system that corresponds to the integral procedure. e input stream is scaled by dt and passed through an adder, whose output is passed back through the same adder. e self-reference in the deﬁnition of int is reﬂected in the ﬁgure by the feedback loop that connects the output of the adder to one of the inputs. Exercise 3.73: We can model electrical circuits using streams to represent the values of currents or voltages at a sequence of times. For instance, suppose we have an RC circuit con- sisting of a resistor of resistance R and a capacitor of capac- itance C in series. e voltage response v of the circuit to an injected current i is determined by the formula in Fig- ure 3.33, whose structure is shown by the accompanying signal-ﬂow diagram. Write a procedure RC that models this circuit. RC should take as inputs the values of R, C, and dt and should return a procedure that takes as inputs a stream representing the current i and an initial value for the capacitor voltage v 0 466 + v -- i R scale: R C v add i 1 scale: integral C Z t v = v0 + 1 i dt + R i v0 C 0 Figure 3.33: An RC circuit and the associated signal-ﬂow diagram. and produces as output the stream of voltages v. For ex- ample, you should be able to use RC to model an RC circuit with R = 5 ohms, C = 1 farad, and a 0.5-second time step by evaluating (define RC1 (RC 5 1 0.5)). is deﬁnes RC1 as a procedure that takes a stream representing the time sequence of currents and an initial capacitor voltage and produces the output stream of voltages. Exercise 3.74: Alyssa P. Hacker is designing a system to process signals coming from physical sensors. One impor- tant feature she wishes to produce is a signal that describes the zero crossings of the input signal. at is, the resulting signal should be +1 whenever the input signal changes from negative to positive, −1 whenever the input signal changes from positive to negative, and 0 otherwise. (Assume that the sign of a 0 input is positive.) For example, a typical in- 467 put signal with its associated zero-crossing signal would be . . . 1 2 1.5 1 0.5 -0.1 -2 -3 -2 -0.5 0.2 3 4 . . . . . . 0 0 0 0 0 -1 0 0 0 0 1 0 0 . . . In Alyssa’s system, the signal from the sensor is represented as a stream sense-data and the stream zero-crossings is the corresponding stream of zero crossings. Alyssa ﬁrst writes a procedure sign-change-detector that takes two values as arguments and compares the signs of the values to produce an appropriate 0, 1, or - 1. She then constructs her zero-crossing stream as follows: (define (make-zero-crossings input-stream last-value) (cons-stream (sign-change-detector (stream-car input-stream) last-value) (make-zero-crossings (stream-cdr input-stream) (stream-car input-stream)))) (define zero-crossings (make-zero-crossings sense-data 0)) Alyssa’s boss, Eva Lu Ator, walks by and suggests that this program is approximately equivalent to the following one, which uses the generalized version of stream-map from Ex- ercise 3.50: (define zero-crossings (stream-map sign-change-detector sense-data ⟨expression⟩)) 468 Complete the program by supplying the indicated ⟨expression⟩. Exercise 3.75: Unfortunately, Alyssa’s zero-crossing de- tector in Exercise 3.74 proves to be insuﬃcient, because the noisy signal from the sensor leads to spurious zero cross- ings. Lem E. Tweakit, a hardware specialist, suggests that Alyssa smooth the signal to ﬁlter out the noise before ex- tracting the zero crossings. Alyssa takes his advice and de- cides to extract the zero crossings from the signal constructed by averaging each value of the sense data with the previous value. She explains the problem to her assistant, Louis Rea- soner, who aempts to implement the idea, altering Alyssa’s program as follows: (define (make-zero-crossings input-stream last-value) (let ((avpt (/ (+ (stream-car input-stream) last-value) 2))) (cons-stream (sign-change-detector avpt last-value) (make-zero-crossings (stream-cdr input-stream) avpt)))) is does not correctly implement Alyssa’s plan. Find the bug that Louis has installed and ﬁx it without changing the structure of the program. (Hint: You will need to increase the number of arguments to make-zero-crossings.) Exercise 3.76: Eva Lu Ator has a criticism of Louis’s ap- proach in Exercise 3.75. e program he wrote is not mod- ular, because it intermixes the operation of smoothing with the zero-crossing extraction. For example, the extractor should 469 not have to be changed if Alyssa ﬁnds a beer way to con- dition her input signal. Help Louis by writing a procedure smooth that takes a stream as input and produces a stream in which each element is the average of two successive in- put stream elements. en use smooth as a component to implement the zero-crossing detector in a more modular style. 3.5.4 Streams and Delayed Evaluation e integral procedure at the end of the preceding section shows how we can use streams to model signal-processing systems that contain feedback loops. e feedback loop for the adder shown in Figure 3.32 is modeled by the fact that integral’s internal stream int is deﬁned in terms of itself: (define int (cons-stream initial-value (add-streams (scale-stream integrand dt) int))) e interpreter’s ability to deal with such an implicit deﬁnition depends on the delay that is incorporated into cons-stream. Without this delay, the interpreter could not construct int before evaluating both argu- ments to cons-stream, which would require that int already be deﬁned. In general, delay is crucial for using streams to model signal-processing systems that contain loops. Without delay, our models would have to be formulated so that the inputs to any signal-processing component would be fully evaluated before the output could be produced. is would outlaw loops. 470 y0 dy y map: f integral Figure 3.34: An “analog computer circuit” that solves the equation dy/dt = f (y). Unfortunately, stream models of systems with loops may require uses of delay beyond the “hidden” delay supplied by cons-stream. For instance, Figure 3.34 shows a signal-processing system for solving the diﬀerential equation dy/dt = f (y) where f is a given function. e ﬁg- ure shows a mapping component, which applies f to its input signal, linked in a feedback loop to an integrator in a manner very similar to that of the analog computer circuits that are actually used to solve such equations. Assuming we are given an initial value y0 for y, we could try to model this system using the procedure (define (solve f y0 dt) (define y (integral dy y0 dt)) (define dy (stream-map f y)) y) is procedure does not work, because in the ﬁrst line of solve the call to integral requires that the input dy be deﬁned, which does not happen until the second line of solve. On the other hand, the intent of our deﬁnition does make sense, because we can, in principle, begin to generate the y stream without 471 knowing dy. Indeed, integral and many other stream operations have properties similar to those of cons-stream, in that we can generate part of the answer given only partial information about the arguments. For integral, the ﬁrst element of the output stream is the speciﬁed initial-value. us, we can generate the ﬁrst element of the output stream without evaluating the integrand dy. Once we know the ﬁrst element of y, the stream-map in the second line of solve can begin working to generate the ﬁrst element of dy, which will produce the next element of y, and so on. To take advantage of this idea, we will redeﬁne integral to expect the integrand stream to be a delayed argument. integral will force the integrand to be evaluated only when it is required to generate more than the ﬁrst element of the output stream: (define (integral delayed-integrand initial-value dt) (define int (cons-stream initial-value (let ((integrand (force delayed-integrand))) (add-streams (scale-stream integrand dt) int)))) int) Now we can implement our solve procedure by delaying the evaluation of dy in the deﬁnition of y:71 (define (solve f y0 dt) (define y (integral (delay dy) y0 dt)) (define dy (stream-map f y)) y) 71 is procedure is not guaranteed to work in all Scheme implementations, although for any implementation there is a simple variation that will work. e problem has to do with subtle diﬀerences in the ways that Scheme implementations handle internal deﬁnitions. (See Section 4.1.6.) 472 In general, every caller of integral must now delay the integrand ar- gument. We can demonstrate that the solve procedure works by ap- proximating e ≈ 2.718 by computing the value at y = 1 of the solution to the diﬀerential equation dy/dt = y with initial condition y(0) = 1: (stream-ref (solve (lambda (y) y) 1 0.001) 1000) 2.716924 Exercise 3.77: e integral procedure used above was analogous to the “implicit” deﬁnition of the inﬁnite stream of integers in Section 3.5.2. Alternatively, we can give a def- inition of integral that is more like integers-starting- from (also in Section 3.5.2): (define (integral integrand initial-value dt) (cons-stream initial-value (if (stream-null? integrand) the-empty-stream (integral (stream-cdr integrand) (+ (* dt (stream-car integrand)) initial-value) dt)))) When used in systems with loops, this procedure has the same problem as does our original version of integral. Modify the procedure so that it expects the integrand as a delayed argument and hence can be used in the solve procedure shown above. 473 dy0 y0 ddy dy y integral integral scale: a add scale: b Figure 3.35: Signal-ﬂow diagram for the solution to a second-order linear diﬀerential equation. Exercise 3.78: Consider the problem of designing a signal- processing system to study the homogeneous second-order linear diﬀerential equation d 2y dy 2 − a − by = 0. dt dt e output stream, modeling y, is generated by a network that contains a loop. is is because the value of d 2y/dt 2 de- pends upon the values of y and dy/dt and both of these are determined by integrating d 2y/dt 2 . e diagram we would like to encode is shown in Figure 3.35. Write a procedure solve-2nd that takes as arguments the constants a, b, and dt and the initial values y0 and dy0 for y and dy/dt and gen- 474 + vR -- iR iL R + iC + vC C L vL -- -- Figure 3.36: A series RLC circuit. erates the stream of successive values of y. Exercise 3.79: Generalize the solve-2nd procedure of Ex- ercise 3.78 so that it can be used to solve general second- order diﬀerential equations d 2y/dt 2 = f (dy/dt , y). Exercise 3.80: A series RLC circuit consists of a resistor, a capacitor, and an inductor connected in series, as shown in Figure 3.36. If R, L, and C are the resistance, inductance, and capacitance, then the relations between voltage (v) and current (i) for the three components are described by the equations di L dvC v R = i R R, vL = L , iC = C , dt dt and the circuit connections dictate the relations i R = i L = −i C , vC = v L + v R . Combining these equations shows that the state of the cir- cuit (summarized by vC , the voltage across the capacitor, 475 scale: 1/L vC integral dvC v C0 scale:-1/C diL iL add integral iL0 scale:-R/L Figure 3.37: A signal-ﬂow diagram for the solution to a series RLC circuit. and i L , the current in the inductor) is described by the pair of diﬀerential equations dvC iL di L 1 R =− , = vC − i L . dt C dt L L e signal-ﬂow diagram representing this system of diﬀer- ential equations is shown in Figure 3.37. Write a procedure RLC that takes as arguments the param- eters R, L, and C of the circuit and the time increment dt. 476 In a manner similar to that of the RC procedure of Exercise 3.73, RLC should produce a procedure that takes the initial values of the state variables, vC 0 and i L 0 , and produces a pair (using cons) of the streams of states vC and i L . Using RLC, generate the pair of streams that models the behavior of a series RLC circuit with R = 1 ohm, C = 0.2 farad, L = 1 henry, dt = 0.1 second, and initial values i L 0 = 0 amps and vC 0 = 10 volts. Normal-order evaluation e examples in this section illustrate how the explicit use of delay and force provides great programming ﬂexibility, but the same exam- ples also show how this can make our programs more complex. Our new integral procedure, for instance, gives us the power to model systems with loops, but we must now remember that integral should be called with a delayed integrand, and every procedure that uses integral must be aware of this. In eﬀect, we have created two classes of procedures: ordinary procedures and procedures that take delayed arguments. In general, creating separate classes of procedures forces us to create sep- arate classes of higher-order procedures as well.72 72 is is a small reﬂection, in Lisp, of the diﬃculties that conventional strongly typed languages such as Pascal have in coping with higher-order procedures. In such lan- guages, the programmer must specify the data types of the arguments and the result of each procedure: number, logical value, sequence, and so on. Consequently, we could not express an abstraction such as “map a given procedure proc over all the elements in a sequence” by a single higher-order procedure such as stream-map. Rather, we would need a diﬀerent mapping procedure for each diﬀerent combination of argument and result data types that might be speciﬁed for a proc. Maintaining a practical notion of “data type” in the presence of higher-order procedures raises many diﬃcult issues. One way of dealing with this problem is illustrated by the language ML (Gordon et al. 1979), 477 One way to avoid the need for two diﬀerent classes of procedures is to make all procedures take delayed arguments. We could adopt a model of evaluation in which all arguments to procedures are automatically delayed and arguments are forced only when they are actually needed (for example, when they are required by a primitive operation). is would transform our language to use normal-order evaluation, which we ﬁrst described when we introduced the substitution model for evalu- ation in Section 1.1.5. Converting to normal-order evaluation provides a uniform and elegant way to simplify the use of delayed evaluation, and this would be a natural strategy to adopt if we were concerned only with stream processing. In Section 4.2, aer we have studied the eval- uator, we will see how to transform our language in just this way. Un- fortunately, including delays in procedure calls wreaks havoc with our ability to design programs that depend on the order of events, such as programs that use assignment, mutate data, or perform input or output. Even the single delay in cons-stream can cause great confusion, as illustrated by Exercise 3.51 and Exercise 3.52. As far as anyone knows, mutability and delayed evaluation do not mix well in programming lan- guages, and devising ways to deal with both of these at once is an active area of research. whose “polymorphic data types” include templates for higher-order transformations between data types. Moreover, data types for most procedures in ML are never explic- itly declared by the programmer. Instead, ML includes a type-inferencing mechanism that uses information in the environment to deduce the data types for newly deﬁned procedures. 478 3.5.5 Modularity of Functional Programs and Modularity of Objects As we saw in Section 3.1.2, one of the major beneﬁts of introducing assignment is that we can increase the modularity of our systems by encapsulating, or “hiding,” parts of the state of a large system within local variables. Stream models can provide an equivalent modularity without the use of assignment. As an illustration, we can reimplement the Monte Carlo estimation of π , which we examined in Section 3.1.2, from a stream-processing point of view. e key modularity issue was that we wished to hide the internal state of a random-number generator from programs that used random numbers. We began with a procedure rand-update, whose successive values furnished our supply of random numbers, and used this to pro- duce a random-number generator: (define rand (let ((x random-init)) (lambda () (set! x (rand-update x)) x))) In the stream formulation there is no random-number generator per se, just a stream of random numbers produced by successive calls to rand- update: (define random-numbers (cons-stream random-init (stream-map rand-update random-numbers))) We use this to construct the stream of outcomes of the Cesàro experi- ment performed on consecutive pairs in the random-numbers stream: 479 (define cesaro-stream (map-successive-pairs (lambda (r1 r2) (= (gcd r1 r2) 1)) random-numbers)) (define (map-successive-pairs f s) (cons-stream (f (stream-car s) (stream-car (stream-cdr s))) (map-successive-pairs f (stream-cdr (stream-cdr s))))) e cesaro-stream is now fed to a monte-carlo procedure, which pro- duces a stream of estimates of probabilities. e results are then con- verted into a stream of estimates of π . is version of the program doesn’t need a parameter telling how many trials to perform. Beer esti- mates of π (from performing more experiments) are obtained by looking farther into the pi stream: (define (monte-carlo experiment-stream passed failed) (define (next passed failed) (cons-stream (/ passed (+ passed failed)) (monte-carlo (stream-cdr experiment-stream) passed failed))) (if (stream-car experiment-stream) (next (+ passed 1) failed) (next passed (+ failed 1)))) (define pi (stream-map (lambda (p) (sqrt (/ 6 p))) (monte-carlo cesaro-stream 0 0))) ere is considerable modularity in this approach, because we still can formulate a general monte-carlo procedure that can deal with arbitrary experiments. Yet there is no assignment or local state. 480 Exercise 3.81: Exercise 3.6 discussed generalizing the random- number generator to allow one to reset the random-number sequence so as to produce repeatable sequences of “ran- dom” numbers. Produce a stream formulation of this same generator that operates on an input stream of requests to generate a new random number or to reset the sequence to a speciﬁed value and that produces the desired stream of random numbers. Don’t use assignment in your solution. Exercise 3.82: Redo Exercise 3.5 on Monte Carlo integra- tion in terms of streams. e stream version of estimate- integral will not have an argument telling how many tri- als to perform. Instead, it will produce a stream of estimates based on successively more trials. A functional-programming view of time Let us now return to the issues of objects and state that were raised at the beginning of this chapter and examine them in a new light. We in- troduced assignment and mutable objects to provide a mechanism for modular construction of programs that model systems with state. We constructed computational objects with local state variables and used assignment to modify these variables. We modeled the temporal behav- ior of the objects in the world by the temporal behavior of the corre- sponding computational objects. Now we have seen that streams provide an alternative way to model objects with local state. We can model a changing quantity, such as the local state of some object, using a stream that represents the time his- tory of successive states. In essence, we represent time explicitly, using streams, so that we decouple time in our simulated world from the se- 481 quence of events that take place during evaluation. Indeed, because of the presence of delay there may be lile relation between simulated time in the model and the order of events during the evaluation. In order to contrast these two approaches to modeling, let us recon- sider the implementation of a “withdrawal processor” that monitors the balance in a bank account. In Section 3.1.3 we implemented a simpliﬁed version of such a processor: (define (make-simplified-withdraw balance) (lambda (amount) (set! balance (- balance amount)) balance)) Calls to make-simplified-withdraw produce computational objects, each with a local state variable balance that is decremented by suc- cessive calls to the object. e object takes an amount as an argument and returns the new balance. We can imagine the user of a bank ac- count typing a sequence of inputs to such an object and observing the sequence of returned values shown on a display screen. Alternatively, we can model a withdrawal processor as a procedure that takes as input a balance and a stream of amounts to withdraw and produces the stream of successive balances in the account: (define (stream-withdraw balance amount-stream) (cons-stream balance (stream-withdraw (- balance (stream-car amount-stream)) (stream-cdr amount-stream)))) stream-withdraw implements a well-deﬁned mathematical function whose output is fully determined by its input. Suppose, however, that the in- put amount-stream is the stream of successive values typed by the user and that the resulting stream of balances is displayed. en, from the 482 perspective of the user who is typing values and watching results, the stream process has the same behavior as the object created by make- simplified-withdraw. However, with the stream version, there is no assignment, no local state variable, and consequently none of the theo- retical diﬃculties that we encountered in Section 3.1.3. Yet the system has state! is is really remarkable. Even though stream-withdraw implements a well-deﬁned mathematical function whose behavior does not change, the user’s perception here is one of interacting with a system that has a changing state. One way to resolve this paradox is to realize that it is the user’s temporal existence that imposes state on the system. If the user could step back from the interaction and think in terms of streams of balances rather than individual transactions, the system would appear stateless.73 From the point of view of one part of a complex process, the other parts appear to change with time. ey have hidden time-varying lo- cal state. If we wish to write programs that model this kind of natural decomposition in our world (as we see it from our viewpoint as a part of that world) with structures in our computer, we make computational objects that are not functional—they must change with time. We model state with local state variables, and we model the changes of state with assignments to those variables. By doing this we make the time of ex- ecution of a computation model time in the world that we are part of, and thus we get “objects” in our computer. Modeling with objects is powerful and intuitive, largely because this matches the perception of interacting with a world of which we are 73 Similarly in physics, when we observe a moving particle, we say that the position (state) of the particle is changing. However, from the perspective of the particle’s world line in space-time there is no change involved. 483 part. However, as we’ve seen repeatedly throughout this chapter, these models raise thorny problems of constraining the order of events and of synchronizing multiple processes. e possibility of avoiding these problems has stimulated the development of functional programming languages, which do not include any provision for assignment or mu- table data. In such a language, all procedures implement well-deﬁned mathematical functions of their arguments, whose behavior does not change. e functional approach is extremely aractive for dealing with concurrent systems.74 On the other hand, if we look closely, we can see time-related prob- lems creeping into functional models as well. One particularly trou- blesome area arises when we wish to design interactive systems, es- pecially ones that model interactions between independent entities. For instance, consider once more the implementation a banking system that permits joint bank accounts. In a conventional system using assignment and objects, we would model the fact that Peter and Paul share an ac- count by having both Peter and Paul send their transaction requests to the same bank-account object, as we saw in Section 3.1.3. From the stream point of view, where there are no “objects” per se, we have al- ready indicated that a bank account can be modeled as a process that operates on a stream of transaction requests to produce a stream of responses. Accordingly, we could model the fact that Peter and Paul have a joint bank account by merging Peter’s stream of transaction re- quests with Paul’s stream of requests and feeding the result to the bank- account stream process, as shown in Figure 3.38. 74 John Backus, the inventor of Fortran, gave high visibility to functional program- ming when he was awarded the Turing award in 1978. His acceptance speech (Backus 1978) strongly advocated the functional approach. A good overview of func- tional programming is given in Henderson 1980 and in Darlington et al. 1982. 484 Peter's requests bank merge Paul's requests account Figure 3.38: A joint bank account, modeled by merging two streams of transaction requests. e trouble with this formulation is in the notion of merge. It will not do to merge the two streams by simply taking alternately one re- quest from Peter and one request from Paul. Suppose Paul accesses the account only very rarely. We could hardly force Peter to wait for Paul to access the account before he could issue a second transaction. However such a merge is implemented, it must interleave the two transaction streams in some way that is constrained by “real time” as perceived by Peter and Paul, in the sense that, if Peter and Paul meet, they can agree that certain transactions were processed before the meeting, and other transactions were processed aer the meeting.75 is is precisely the same constraint that we had to deal with in Section 3.4.1, where we found the need to introduce explicit synchronization to ensure a “correct” order of events in concurrent processing of objects with state. us, in an aempt to support the functional style, the need to merge inputs from diﬀerent agents reintroduces the same problems that the functional style was meant to eliminate. 75 Observe that, for any two streams, there is in general more than one acceptable or- der of interleaving. us, technically, “merge” is a relation rather than a function—the answer is not a deterministic function of the inputs. We already mentioned (Footnote 39) that nondeterminism is essential when dealing with concurrency. e merge rela- tion illustrates the same essential nondeterminism, from the functional perspective. In Section 4.3, we will look at nondeterminism from yet another point of view. 485 We began this chapter with the goal of building computational mod- els whose structure matches our perception of the real world we are trying to model. We can model the world as a collection of separate, time-bound, interacting objects with state, or we can model the world as a single, timeless, stateless unity. Each view has powerful advantages, but neither view alone is completely satisfactory. A grand uniﬁcation has yet to emerge.76 76 e object model approximates the world by dividing it into separate pieces. e functional model does not modularize along object boundaries. e object model is useful when the unshared state of the “objects” is much larger than the state that they share. An example of a place where the object viewpoint fails is quantum mechanics, where thinking of things as individual particles leads to paradoxes and confusions. Uni- fying the object view with the functional view may have lile to do with programming, but rather with fundamental epistemological issues. 486 Metalinguistic Abstraction . . . It’s in words that the magic is—Abracadabra, Open Sesame, and the rest—but the magic words in one story aren’t magi- cal in the next. e real magic is to understand which words work, and when, and for what; the trick is to learn the trick. . . . And those words are made from the leers of our alpha- bet: a couple-dozen squiggles we can draw with the pen. is is the key! And the treasure, too, if we can only get our hands on it! It’s as if—as if the key to the treasure is the treasure! —John Barth, Chimera I , we have seen that expert program- mers control the complexity of their designs with the same general techniques used by designers of all complex systems. ey combine primitive elements to form compound objects, they abstract compound 487 objects to form higher-level building blocks, and they preserve modu- larity by adopting appropriate large-scale views of system structure. In illustrating these techniques, we have used Lisp as a language for de- scribing processes and for constructing computational data objects and processes to model complex phenomena in the real world. However, as we confront increasingly complex problems, we will ﬁnd that Lisp, or indeed any ﬁxed programming language, is not suﬃcient for our needs. We must constantly turn to new languages in order to express our ideas more eﬀectively. Establishing new languages is a powerful strategy for controlling complexity in engineering design; we can oen enhance our ability to deal with a complex problem by adopting a new language that enables us to describe (and hence to think about) the problem in a dif- ferent way, using primitives, means of combination, and means of ab- straction that are particularly well suited to the problem at hand.1 Programming is endowed with a multitude of languages. ere are 1 e same idea is pervasive throughout all of engineering. For example, electri- cal engineers use many diﬀerent languages for describing circuits. Two of these are the language of electrical networks and the language of electrical systems. e network language emphasizes the physical modeling of devices in terms of discrete electrical el- ements. e primitive objects of the network language are primitive electrical compo- nents such as resistors, capacitors, inductors, and transistors, which are characterized in terms of physical variables called voltage and current. When describing circuits in the network language, the engineer is concerned with the physical characteristics of a design. In contrast, the primitive objects of the system language are signal-processing modules such as ﬁlters and ampliﬁers. Only the functional behavior of the modules is relevant, and signals are manipulated without concern for their physical realization as voltages and currents. e system language is erected on the network language, in the sense that the elements of signal-processing systems are constructed from electrical networks. Here, however, the concerns are with the large-scale organization of elec- trical devices to solve a given application problem; the physical feasibility of the parts is assumed. is layered collection of languages is another example of the stratiﬁed design technique illustrated by the picture language of Section 2.2.4. 488 physical languages, such as the machine languages for particular com- puters. ese languages are concerned with the representation of data and control in terms of individual bits of storage and primitive machine instructions. e machine-language programmer is concerned with us- ing the given hardware to erect systems and utilities for the eﬃcient im- plementation of resource-limited computations. High-level languages, erected on a machine-language substrate, hide concerns about the rep- resentation of data as collections of bits and the representation of pro- grams as sequences of primitive instructions. ese languages have means of combination and abstraction, such as procedure deﬁnition, that are appropriate to the larger-scale organization of systems. Metalinguistic abstraction—establishing new languages—plays an im- portant role in all branches of engineering design. It is particularly im- portant to computer programming, because in programming not only can we formulate new languages but we can also implement these lan- guages by constructing evaluators. An evaluator (or interpreter ) for a programming language is a procedure that, when applied to an expres- sion of the language, performs the actions required to evaluate that ex- pression. It is no exaggeration to regard this as the most fundamental idea in programming: e evaluator, which determines the meaning of expres- sions in a programming language, is just another program. To appreciate this point is to change our images of ourselves as pro- grammers. We come to see ourselves as designers of languages, rather than only users of languages designed by others. In fact, we can regard almost any program as the evaluator for some language. For instance, the polynomial manipulation system of Section 489 2.5.3 embodies the rules of polynomial arithmetic and implements them in terms of operations on list-structured data. If we augment this system with procedures to read and print polynomial expressions, we have the core of a special-purpose language for dealing with problems in sym- bolic mathematics. e digital-logic simulator of Section 3.3.4 and the constraint propagator of Section 3.3.5 are legitimate languages in their own right, each with its own primitives, means of combination, and means of abstraction. Seen from this perspective, the technology for coping with large-scale computer systems merges with the technology for building new computer languages, and computer science itself be- comes no more (and no less) than the discipline of constructing appro- priate descriptive languages. We now embark on a tour of the technology by which languages are established in terms of other languages. In this chapter we shall use Lisp as a base, implementing evaluators as Lisp procedures. Lisp is particu- larly well suited to this task, because of its ability to represent and ma- nipulate symbolic expressions. We will take the ﬁrst step in understand- ing how languages are implemented by building an evaluator for Lisp itself. e language implemented by our evaluator will be a subset of the Scheme dialect of Lisp that we use in this book. Although the evaluator described in this chapter is wrien for a particular dialect of Lisp, it con- tains the essential structure of an evaluator for any expression-oriented language designed for writing programs for a sequential machine. (In fact, most language processors contain, deep within them, a lile “Lisp” evaluator.) e evaluator has been simpliﬁed for the purposes of illus- tration and discussion, and some features have been le out that would be important to include in a production-quality Lisp system. Neverthe- less, this simple evaluator is adequate to execute most of the programs 490 in this book.2 An important advantage of making the evaluator accessible as a Lisp program is that we can implement alternative evaluation rules by describing these as modiﬁcations to the evaluator program. One place where we can use this power to good eﬀect is to gain extra control over the ways in which computational models embody the notion of time, which was so central to the discussion in Chapter 3. ere, we mitigated some of the complexities of state and assignment by using streams to decouple the representation of time in the world from time in the com- puter. Our stream programs, however, were sometimes cumbersome, because they were constrained by the applicative-order evaluation of Scheme. In Section 4.2, we’ll change the underlying language to provide for a more elegant approach, by modifying the evaluator to provide for normal-order evaluation. Section 4.3 implements a more ambitious linguistic change, whereby expressions have many values, rather than just a single value. In this language of nondeterministic computing, it is natural to express processes that generate all possible values for expressions and then search for those values that satisfy certain constraints. In terms of models of com- putation and time, this is like having time branch into a set of “possible futures” and then searching for appropriate time lines. With our nonde- terministic evaluator, keeping track of multiple values and performing searches are handled automatically by the underlying mechanism of the language. In Section 4.4 we implement a logic-programming language in which 2 e most important features that our evaluator leaves out are mechanisms for han- dling errors and supporting debugging. For a more extensive discussion of evaluators, see Friedman et al. 1992, which gives an exposition of programming languages that proceeds via a sequence of evaluators wrien in Scheme. 491 knowledge is expressed in terms of relations, rather than in terms of computations with inputs and outputs. Even though this makes the lan- guage drastically diﬀerent from Lisp, or indeed from any conventional language, we will see that the logic-programming evaluator shares the essential structure of the Lisp evaluator. 4.1 The Metacircular Evaluator Our evaluator for Lisp will be implemented as a Lisp program. It may seem circular to think about evaluating Lisp programs using an evalua- tor that is itself implemented in Lisp. However, evaluation is a process, so it is appropriate to describe the evaluation process using Lisp, which, aer all, is our tool for describing processes.3 An evaluator that is writ- ten in the same language that it evaluates is said to be metacircular. e metacircular evaluator is essentially a Scheme formulation of the environment model of evaluation described in Section 3.2. Recall that the model has two basic parts: 1. To evaluate a combination (a compound expression other than a special form), evaluate the subexpressions and then apply the value of the operator subexpression to the values of the operand subexpressions. 2. To apply a compound procedure to a set of arguments, evaluate the body of the procedure in a new environment. To construct 3 Even so, there will remain important aspects of the evaluation process that are not elucidated by our evaluator. e most important of these are the detailed mechanisms by which procedures call other procedures and return values to their callers. We will address these issues in Chapter 5, where we take a closer look at the evaluation process by implementing the evaluator as a simple register machine. 492 this environment, extend the environment part of the procedure object by a frame in which the formal parameters of the procedure are bound to the arguments to which the procedure is applied. ese two rules describe the essence of the evaluation process, a basic cycle in which expressions to be evaluated in environments are reduced to procedures to be applied to arguments, which in turn are reduced to new expressions to be evaluated in new environments, and so on, un- til we get down to symbols, whose values are looked up in the envi- ronment, and to primitive procedures, which are applied directly (see Figure 4.1).4 is evaluation cycle will be embodied by the interplay between the two critical procedures in the evaluator, eval and apply, which are described in Section 4.1.1 (see Figure 4.1). e implementation of the evaluator will depend upon procedures 4 If we grant ourselves the ability to apply primitives, then what remains for us to implement in the evaluator? e job of the evaluator is not to specify the primitives of the language, but rather to provide the connective tissue—the means of combination and the means of abstraction—that binds a collection of primitives to form a language. Speciﬁcally: • e evaluator enables us to deal with nested expressions. For example, although simply applying primitives would suﬃce for evaluating the expression (+ 1 6), it is not adequate for handling (+ 1 (* 2 3)). As far as the primitive procedure + is concerned, its arguments must be numbers, and it would choke if we passed it the expression (* 2 3) as an argument. One important role of the evaluator is to choreograph procedure composition so that (* 2 3) is reduced to 6 before being passed as an argument to +. • e evaluator allows us to use variables. For example, the primitive procedure for addition has no way to deal with expressions such as (+ x 1). We need an evaluator to keep track of variables and obtain their values before invoking the primitive procedures. • e evaluator allows us to deﬁne compound procedures. is involves keeping track of procedure deﬁnitions, knowing how to use these deﬁnitions in evaluating ex- pressions, and providing a mechanism that enables procedures to accept arguments. • e evaluator provides the special forms, which must be evaluated diﬀerently from procedure calls. 493 Procedure, Expression, Eval Apply Arguments Environment Figure 4.1: e eval-apply cycle exposes the essence of a computer language. that deﬁne the syntax of the expressions to be evaluated. We will use data abstraction to make the evaluator independent of the representa- tion of the language. For example, rather than commiing to a choice that an assignment is to be represented by a list beginning with the symbol set! we use an abstract predicate assignment? to test for an assignment, and we use abstract selectors assignment-variable and assignment-value to access the parts of an assignment. Implementa- tion of expressions will be described in detail in Section 4.1.2. ere are also operations, described in Section 4.1.3, that specify the represen- tation of procedures and environments. For example, make-procedure constructs compound procedures, lookup-variable-value accesses the values of variables, and apply-primitive-procedure applies a primi- tive procedure to a given list of arguments. 494 4.1.1 The Core of the Evaluator e evaluation process can be described as the interplay between two procedures: eval and apply. Eval eval takes as arguments an expression and an environment. It classi- ﬁes the expression and directs its evaluation. eval is structured as a case analysis of the syntactic type of the expression to be evaluated. In or- der to keep the procedure general, we express the determination of the type of an expression abstractly, making no commitment to any partic- ular representation for the various types of expressions. Each type of expression has a predicate that tests for it and an abstract means for selecting its parts. is abstract syntax makes it easy to see how we can change the syntax of the language by using the same evaluator, but with a diﬀerent collection of syntax procedures. Primitive expressions • For self-evaluating expressions, such as numbers, eval returns the expression itself. • eval must look up variables in the environment to ﬁnd their val- ues. Special forms • For quoted expressions, eval returns the expression that was quoted. • An assignment to (or a deﬁnition o) a variable must recursively call eval to compute the new value to be associated with the vari- able. e environment must be modiﬁed to change (or create) the binding of the variable. 495 • An if expression requires special processing of its parts, so as to evaluate the consequent if the predicate is true, and otherwise to evaluate the alternative. • A lambda expression must be transformed into an applicable pro- cedure by packaging together the parameters and body speciﬁed by the lambda expression with the environment of the evaluation. • A begin expression requires evaluating its sequence of expres- sions in the order in which they appear. • A case analysis (cond) is transformed into a nest of if expressions and then evaluated. Combinations • For a procedure application, eval must recursively evaluate the operator part and the operands of the combination. e resulting procedure and arguments are passed to apply, which handles the actual procedure application. Here is the deﬁnition of eval: (define (eval exp env) (cond ((self-evaluating? exp) exp) ((variable? exp) (lookup-variable-value exp env)) ((quoted? exp) (text-of-quotation exp)) ((assignment? exp) (eval-assignment exp env)) ((definition? exp) (eval-definition exp env)) ((if? exp) (eval-if exp env)) ((lambda? exp) (make-procedure (lambda-parameters exp) (lambda-body exp) env)) 496 ((begin? exp) (eval-sequence (begin-actions exp) env)) ((cond? exp) (eval (cond->if exp) env)) ((application? exp) (apply (eval (operator exp) env) (list-of-values (operands exp) env))) (else (error "Unknown expression type: EVAL" exp)))) For clarity, eval has been implemented as a case analysis using cond. e disadvantage of this is that our procedure handles only a few distin- guishable types of expressions, and no new ones can be deﬁned without editing the deﬁnition of eval. In most Lisp implementations, dispatch- ing on the type of an expression is done in a data-directed style. is allows a user to add new types of expressions that eval can distinguish, without modifying the deﬁnition of eval itself. (See Exercise 4.3.) Apply apply takes two arguments, a procedure and a list of arguments to which the procedure should be applied. apply classiﬁes procedures into two kinds: It calls apply-primitive-procedure to apply primitives; it applies compound procedures by sequentially evaluating the expres- sions that make up the body of the procedure. e environment for the evaluation of the body of a compound procedure is constructed by extending the base environment carried by the procedure to include a frame that binds the parameters of the procedure to the arguments to which the procedure is to be applied. Here is the deﬁnition of apply: (define (apply procedure arguments) (cond ((primitive-procedure? procedure) (apply-primitive-procedure procedure arguments)) 497 ((compound-procedure? procedure) (eval-sequence (procedure-body procedure) (extend-environment (procedure-parameters procedure) arguments (procedure-environment procedure)))) (else (error "Unknown procedure type: APPLY" procedure)))) Procedure arguments When eval processes a procedure application, it uses list-of-values to produce the list of arguments to which the procedure is to be applied. list-of-values takes as an argument the operands of the combina- tion. It evaluates each operand and returns a list of the corresponding values:5 (define (list-of-values exps env) (if (no-operands? exps) '() (cons (eval (first-operand exps) env) (list-of-values (rest-operands exps) env)))) 5 We could have simpliﬁed the application? clause in eval by using map (and stip- ulating that operands returns a list) rather than writing an explicit list-of-values procedure. We chose not to use map here to emphasize the fact that the evaluator can be implemented without any use of higher-order procedures (and thus could be writ- ten in a language that doesn’t have higher-order procedures), even though the language that it supports will include higher-order procedures. 498 Conditionals eval-if evaluates the predicate part of an if expression in the given environment. If the result is true, eval-if evaluates the consequent, otherwise it evaluates the alternative: (define (eval-if exp env) (if (true? (eval (if-predicate exp) env)) (eval (if-consequent exp) env) (eval (if-alternative exp) env))) e use of true? in eval-if highlights the issue of the connection be- tween an implemented language and an implementation language. e if-predicate is evaluated in the language being implemented and thus yields a value in that language. e interpreter predicate true? trans- lates that value into a value that can be tested by the if in the imple- mentation language: e metacircular representation of truth might not be the same as that of the underlying Scheme.6 Sequences eval-sequence is used by apply to evaluate the sequence of expressions in a procedure body and by eval to evaluate the sequence of expressions in a begin expression. It takes as arguments a sequence of expressions and an environment, and evaluates the expressions in the order in which they occur. e value returned is the value of the ﬁnal expression. (define (eval-sequence exps env) (cond ((last-exp? exps) (eval (first-exp exps) env)) 6 In this case, the language being implemented and the implementation language are the same. Contemplation of the meaning of true? here yields expansion of conscious- ness without the abuse of substance. 499 (else (eval (first-exp exps) env) (eval-sequence (rest-exps exps) env)))) Assignments and definitions e following procedure handles assignments to variables. It calls eval to ﬁnd the value to be assigned and transmits the variable and the re- sulting value to set-variable-value! to be installed in the designated environment. (define (eval-assignment exp env) (set-variable-value! (assignment-variable exp) (eval (assignment-value exp) env) env) 'ok) Deﬁnitions of variables are handled in a similar manner.7 (define (eval-definition exp env) (define-variable! (definition-variable exp) (eval (definition-value exp) env) env) 'ok) We have chosen here to return the symbol ok as the value of an assign- ment or a deﬁnition.8 Exercise 4.1: Notice that we cannot tell whether the metacir- cular evaluator evaluates operands from le to right or from 7 is implementation of define ignores a subtle issue in the handling of internal deﬁnitions, although it works correctly in most cases. We will see what the problem is and how to solve it in Section 4.1.6. 8 As we said when we introduced define and set!, these values are implementation- dependent in Scheme—that is, the implementor can choose what value to return. 500 right to le. Its evaluation order is inherited from the un- derlying Lisp: If the arguments to cons in list-of-values are evaluated from le to right, then list-of-values will evaluate operands from le to right; and if the arguments to cons are evaluated from right to le, then list-of-values will evaluate operands from right to le. Write a version of list-of-values that evaluates operands from le to right regardless of the order of evaluation in the underlying Lisp. Also write a version of list-of-values that evaluates operands from right to le. 4.1.2 Representing Expressions e evaluator is reminiscent of the symbolic diﬀerentiation program discussed in Section 2.3.2. Both programs operate on symbolic expres- sions. In both programs, the result of operating on a compound expres- sion is determined by operating recursively on the pieces of the expres- sion and combining the results in a way that depends on the type of the expression. In both programs we used data abstraction to decouple the general rules of operation from the details of how expressions are represented. In the diﬀerentiation program this meant that the same diﬀerentiation procedure could deal with algebraic expressions in pre- ﬁx form, in inﬁx form, or in some other form. For the evaluator, this means that the syntax of the language being evaluated is determined solely by the procedures that classify and extract pieces of expressions. Here is the speciﬁcation of the syntax of our language: • e only self-evaluating items are numbers and strings: (define (self-evaluating? exp) (cond ((number? exp) true) 501 ((string? exp) true) (else false))) • Variables are represented by symbols: (define (variable? exp) (symbol? exp)) • otations have the form (quote ⟨text-of-quotation⟩):9 (define (quoted? exp) (tagged-list? exp 'quote)) (define (text-of-quotation exp) (cadr exp)) quoted? is deﬁned in terms of the procedure tagged-list?, which identiﬁes lists beginning with a designated symbol: (define (tagged-list? exp tag) (if (pair? exp) (eq? (car exp) tag) false)) • Assignments have the form (set! ⟨var⟩ ⟨value⟩): (define (assignment? exp) (tagged-list? exp 'set!)) (define (assignment-variable exp) (cadr exp)) (define (assignment-value exp) (caddr exp)) • Deﬁnitions have the form (define ⟨var⟩ ⟨value⟩) or the form 9 As mentioned in Section 2.3.1, the evaluator sees a quoted expression as a list begin- ning with quote, even if the expression is typed with the quotation mark. For example, the expression 'a would be seen by the evaluator as (quote a). See Exercise 2.55. 502 (define (⟨var⟩ ⟨parameter1 ⟩ . . . ⟨parametern ⟩) ⟨body⟩) e laer form (standard procedure deﬁnition) is syntactic sugar for (define ⟨var⟩ (lambda (⟨parameter1 ⟩ . . . ⟨parametern ⟩) ⟨body⟩)) e corresponding syntax procedures are the following: (define (definition? exp) (tagged-list? exp 'define)) (define (definition-variable exp) (if (symbol? (cadr exp)) (cadr exp) (caadr exp))) (define (definition-value exp) (if (symbol? (cadr exp)) (caddr exp) (make-lambda (cdadr exp) ; formal parameters (cddr exp)))) ; body • lambda expressions are lists that begin with the symbol lambda: (define (lambda? exp) (tagged-list? exp 'lambda)) (define (lambda-parameters exp) (cadr exp)) (define (lambda-body exp) (cddr exp)) We also provide a constructor for lambda expressions, which is used by definition-value, above: (define (make-lambda parameters body) (cons 'lambda (cons parameters body))) 503 • Conditionals begin with if and have a predicate, a consequent, and an (optional) alternative. If the expression has no alternative part, we provide false as the alternative.10 (define (if? exp) (tagged-list? exp 'if)) (define (if-predicate exp) (cadr exp)) (define (if-consequent exp) (caddr exp)) (define (if-alternative exp) (if (not (null? (cdddr exp))) (cadddr exp) 'false)) We also provide a constructor for if expressions, to be used by cond->if to transform cond expressions into if expressions: (define (make-if predicate consequent alternative) (list 'if predicate consequent alternative)) • begin packages a sequence of expressions into a single expres- sion. We include syntax operations on begin expressions to ex- tract the actual sequence from the begin expression, as well as selectors that return the ﬁrst expression and the rest of the ex- pressions in the sequence.11 (define (begin? exp) (tagged-list? exp 'begin)) (define (begin-actions exp) (cdr exp)) 10 e value of an if expression when the predicate is false and there is no alternative is unspeciﬁed in Scheme; we have chosen here to make it false. We will support the use of the variables true and false in expressions to be evaluated by binding them in the global environment. See Section 4.1.4. 11 ese selectors for a list of expressions—and the corresponding ones for a list of operands—are not intended as a data abstraction. ey are introduced as mnemonic names for the basic list operations in order to make it easier to understand the explicit- control evaluator in Section 5.4. 504 (define (last-exp? seq) (null? (cdr seq))) (define (first-exp seq) (car seq)) (define (rest-exps seq) (cdr seq)) We also include a constructor sequence->exp (for use by cond- >if)that transforms a sequence into a single expression, using begin if necessary: (define (sequence->exp seq) (cond ((null? seq) seq) ((last-exp? seq) (first-exp seq)) (else (make-begin seq)))) (define (make-begin seq) (cons 'begin seq)) • A procedure application is any compound expression that is not one of the above expression types. e car of the expression is the operator, and the cdr is the list of operands: (define (application? exp) (pair? exp)) (define (operator exp) (car exp)) (define (operands exp) (cdr exp)) (define (no-operands? ops) (null? ops)) (define (first-operand ops) (car ops)) (define (rest-operands ops) (cdr ops)) Derived expressions Some special forms in our language can be deﬁned in terms of expres- sions involving other special forms, rather than being implemented di- rectly. One example is cond, which can be implemented as a nest of if expressions. For example, we can reduce the problem of evaluating the expression 505 (cond ((> x 0) x) ((= x 0) (display 'zero) 0) (else (- x))) to the problem of evaluating the following expression involving if and begin expressions: (if (> x 0) x (if (= x 0) (begin (display 'zero) 0) (- x))) Implementing the evaluation of cond in this way simpliﬁes the evalua- tor because it reduces the number of special forms for which the evalu- ation process must be explicitly speciﬁed. We include syntax procedures that extract the parts of a cond ex- pression, and a procedure cond->if that transforms cond expressions into if expressions. A case analysis begins with cond and has a list of predicate-action clauses. A clause is an else clause if its predicate is the symbol else.12 (define (cond? exp) (tagged-list? exp 'cond)) (define (cond-clauses exp) (cdr exp)) (define (cond-else-clause? clause) (eq? (cond-predicate clause) 'else)) (define (cond-predicate clause) (car clause)) (define (cond-actions clause) (cdr clause)) (define (cond->if exp) (expand-clauses (cond-clauses exp))) (define (expand-clauses clauses) (if (null? clauses) 'false ; no else clause 12 e value of a cond expression when all the predicates are false and there is no else clause is unspeciﬁed in Scheme; we have chosen here to make it false. 506 (let ((first (car clauses)) (rest (cdr clauses))) (if (cond-else-clause? first) (if (null? rest) (sequence->exp (cond-actions first)) (error "ELSE clause isn't last: COND->IF" clauses)) (make-if (cond-predicate first) (sequence->exp (cond-actions first)) (expand-clauses rest)))))) Expressions (such as cond) that we choose to implement as syntactic transformations are called derived expressions. let expressions are also derived expressions (see Exercise 4.6).13 Exercise 4.2: Louis Reasoner plans to reorder the cond clauses in eval so that the clause for procedure applications ap- pears before the clause for assignments. He argues that this will make the interpreter more eﬃcient: Since programs usually contain more applications than assignments, def- initions, and so on, his modiﬁed eval will usually check fewer clauses than the original eval before identifying the type of an expression. a. What is wrong with Louis’s plan? (Hint: What will 13 Practical Lisp systems provide a mechanism that allows a user to add new de- rived expressions and specify their implementation as syntactic transformations with- out modifying the evaluator. Such a user-deﬁned transformation is called a macro. Al- though it is easy to add an elementary mechanism for deﬁning macros, the result- ing language has subtle name-conﬂict problems. ere has been much research on mechanisms for macro deﬁnition that do not cause these diﬃculties. See, for example, Kohlbecker 1986, Clinger and Rees 1991, and Hanson 1991. 507 Louis’s evaluator do with the expression (define x 3)?) b. Louis is upset that his plan didn’t work. He is will- ing to go to any lengths to make his evaluator recog- nize procedure applications before it checks for most other kinds of expressions. Help him by changing the syntax of the evaluated language so that procedure applications start with call. For example, instead of (factorial 3) we will now have to write (call factorial 3) and instead of (+ 1 2) we will have to write (call + 1 2). Exercise 4.3: Rewrite eval so that the dispatch is done in data-directed style. Compare this with the data-directed diﬀerentiation procedure of Exercise 2.73. (You may use the car of a compound expression as the type of the expres- sion, as is appropriate for the syntax implemented in this section.) Exercise 4.4: Recall the deﬁnitions of the special forms and and or from Chapter 1: • and: e expressions are evaluated from le to right. If any expression evaluates to false, false is returned; any remaining expressions are not evaluated. If all the expressions evaluate to true values, the value of the last expression is returned. If there are no expressions then true is returned. • or: e expressions are evaluated from le to right. If any expression evaluates to a true value, that value 508 is returned; any remaining expressions are not evalu- ated. If all expressions evaluate to false, or if there are no expressions, then false is returned. Install and and or as new special forms for the evaluator by deﬁning appropriate syntax procedures and evaluation pro- cedures eval-and and eval-or. Alternatively, show how to implement and and or as derived expressions. Exercise 4.5: Scheme allows an additional syntax for cond clauses, (⟨test⟩ => ⟨recipient⟩). If ⟨test ⟩ evaluates to a true value, then ⟨recipient ⟩ is evaluated. Its value must be a procedure of one argument; this procedure is then invoked on the value of the ⟨test ⟩, and the result is returned as the value of the cond expression. For example (cond ((assoc 'b '((a 1) (b 2))) => cadr) (else false)) returns 2. Modify the handling of cond so that it supports this extended syntax. Exercise 4.6: let expressions are derived expressions, be- cause (let ((⟨var1 ⟩ ⟨exp1 ⟩) . . . (⟨varn ⟩ ⟨expn ⟩)) ⟨body⟩) is equivalent to ((lambda (⟨var1 ⟩ . . . ⟨varn ⟩) ⟨body⟩) ⟨exp1 ⟩ ... ⟨expn ⟩) 509 Implement a syntactic transformation let->combination that reduces evaluating let expressions to evaluating com- binations of the type shown above, and add the appropriate clause to eval to handle let expressions. Exercise 4.7: let* is similar to let, except that the bind- ings of the let* variables are performed sequentially from le to right, and each binding is made in an environment in which all of the preceding bindings are visible. For example (let* ((x 3) (y (+ x 2)) (z (+ x y 5))) (* x z)) returns 39. Explain how a let* expression can be rewrien as a set of nested let expressions, and write a procedure let*->nested-lets that performs this transformation. If we have already implemented let (Exercise 4.6) and we want to extend the evaluator to handle let*, is it suﬃcient to add a clause to eval whose action is (eval (let*->nested-lets exp) env) or must we explicitly expand let* in terms of non-derived expressions? Exercise 4.8: “Named let” is a variant of let that has the form (let ⟨var⟩ ⟨bindings⟩ ⟨body⟩) e ⟨bindings ⟩ and ⟨body ⟩ are just as in ordinary let, ex- cept that ⟨var ⟩ is bound within ⟨body ⟩ to a procedure whose body is ⟨body ⟩ and whose parameters are the variables in 510 the ⟨bindings ⟩. us, one can repeatedly execute the ⟨body ⟩ by invoking the procedure named ⟨var ⟩. For example, the iterative Fibonacci procedure (Section 1.2.2) can be rewrit- ten using named let as follows: (define (fib n) (let fib-iter ((a 1) (b 0) (count n)) (if (= count 0) b (fib-iter (+ a b) a (- count 1))))) Modify let->combination of Exercise 4.6 to also support named let. Exercise 4.9: Many languages support a variety of iteration constructs, such as do, for, while, and until. In Scheme, iterative processes can be expressed in terms of ordinary procedure calls, so special iteration constructs provide no essential gain in computational power. On the other hand, such constructs are oen convenient. Design some itera- tion constructs, give examples of their use, and show how to implement them as derived expressions. Exercise 4.10: By using data abstraction, we were able to write an eval procedure that is independent of the particu- lar syntax of the language to be evaluated. To illustrate this, design and implement a new syntax for Scheme by modify- ing the procedures in this section, without changing eval or apply. 511 4.1.3 Evaluator Data Structures In addition to deﬁning the external syntax of expressions, the evaluator implementation must also deﬁne the data structures that the evaluator manipulates internally, as part of the execution of a program, such as the representation of procedures and environments and the representation of true and false. Testing of predicates For conditionals, we accept anything to be true that is not the explicit false object. (define (true? x) (not (eq? x false))) (define (false? x) (eq? x false)) Representing procedures To handle primitives, we assume that we have available the following procedures: • (apply-primitive-procedure ⟨proc⟩ ⟨args⟩) applies the given primitive procedure to the argument values in the list ⟨args ⟩ and returns the result of the application. • (primitive-procedure? ⟨proc⟩) tests whether ⟨proc ⟩ is a primitive procedure. ese mechanisms for handling primitives are further described in Sec- tion 4.1.4. Compound procedures are constructed from parameters, procedure bodies, and environments using the constructor make-procedure: 512 (define (make-procedure parameters body env) (list 'procedure parameters body env)) (define (compound-procedure? p) (tagged-list? p 'procedure)) (define (procedure-parameters p) (cadr p)) (define (procedure-body p) (caddr p)) (define (procedure-environment p) (cadddr p)) Operations on Environments e evaluator needs operations for manipulating environments. As ex- plained in Section 3.2, an environment is a sequence of frames, where each frame is a table of bindings that associate variables with their cor- responding values. We use the following operations for manipulating environments: • (lookup-variable-value ⟨var⟩ ⟨env⟩) returns the value that is bound to the symbol ⟨var ⟩ in the envi- ronment ⟨env ⟩, or signals an error if the variable is unbound. • (extend-environment ⟨variables⟩ ⟨values⟩ ⟨base-env⟩) returns a new environment, consisting of a new frame in which the symbols in the list ⟨variables ⟩ are bound to the corresponding elements in the list ⟨values ⟩, where the enclosing environment is the environment ⟨base-env ⟩. • (define-variable! ⟨var⟩ ⟨value⟩ ⟨env⟩) adds to the ﬁrst frame in the environment ⟨env ⟩ a new binding that associates the variable ⟨var ⟩ with the value ⟨value ⟩. 513 • (set-variable-value! ⟨var⟩ ⟨value⟩ ⟨env⟩) changes the binding of the variable ⟨var ⟩ in the environment ⟨env ⟩ so that the variable is now bound to the value ⟨value ⟩, or signals an error if the variable is unbound. To implement these operations we represent an environment as a list of frames. e enclosing environment of an environment is the cdr of the list. e empty environment is simply the empty list. (define (enclosing-environment env) (cdr env)) (define (first-frame env) (car env)) (define the-empty-environment '()) Each frame of an environment is represented as a pair of lists: a list of the variables bound in that frame and a list of the associated values.14 (define (make-frame variables values) (cons variables values)) (define (frame-variables frame) (car frame)) (define (frame-values frame) (cdr frame)) (define (add-binding-to-frame! var val frame) (set-car! frame (cons var (car frame))) (set-cdr! frame (cons val (cdr frame)))) To extend an environment by a new frame that associates variables with values, we make a frame consisting of the list of variables and the list of values, and we adjoin this to the environment. We signal an error if the number of variables does not match the number of values. 14 Frames are not really a data abstraction in the following code: set-variable- value! and define-variable! use set-car! to directly modify the values in a frame. e purpose of the frame procedures is to make the environment-manipulation proce- dures easy to read. 514 (define (extend-environment vars vals base-env) (if (= (length vars) (length vals)) (cons (make-frame vars vals) base-env) (if (< (length vars) (length vals)) (error "Too many arguments supplied" vars vals) (error "Too few arguments supplied" vars vals)))) To look up a variable in an environment, we scan the list of variables in the ﬁrst frame. If we ﬁnd the desired variable, we return the corre- sponding element in the list of values. If we do not ﬁnd the variable in the current frame, we search the enclosing environment, and so on. If we reach the empty environment, we signal an “unbound variable” error. (define (lookup-variable-value var env) (define (env-loop env) (define (scan vars vals) (cond ((null? vars) (env-loop (enclosing-environment env))) ((eq? var (car vars)) (car vals)) (else (scan (cdr vars) (cdr vals))))) (if (eq? env the-empty-environment) (error "Unbound variable" var) (let ((frame (first-frame env))) (scan (frame-variables frame) (frame-values frame))))) (env-loop env)) To set a variable to a new value in a speciﬁed environment, we scan for the variable, just as in lookup-variable-value, and change the corre- sponding value when we ﬁnd it. (define (set-variable-value! var val env) (define (env-loop env) 515 (define (scan vars vals) (cond ((null? vars) (env-loop (enclosing-environment env))) ((eq? var (car vars)) (set-car! vals val)) (else (scan (cdr vars) (cdr vals))))) (if (eq? env the-empty-environment) (error "Unbound variable: SET!" var) (let ((frame (first-frame env))) (scan (frame-variables frame) (frame-values frame))))) (env-loop env)) To deﬁne a variable, we search the ﬁrst frame for a binding for the variable, and change the binding if it exists (just as in set-variable- value!). If no such binding exists, we adjoin one to the ﬁrst frame. (define (define-variable! var val env) (let ((frame (first-frame env))) (define (scan vars vals) (cond ((null? vars) (add-binding-to-frame! var val frame)) ((eq? var (car vars)) (set-car! vals val)) (else (scan (cdr vars) (cdr vals))))) (scan (frame-variables frame) (frame-values frame)))) e method described here is only one of many plausible ways to rep- resent environments. Since we used data abstraction to isolate the rest of the evaluator from the detailed choice of representation, we could change the environment representation if we wanted to. (See Exercise 4.11.) In a production-quality Lisp system, the speed of the evaluator’s environment operations—especially that of variable lookup—has a ma- jor impact on the performance of the system. e representation de- scribed here, although conceptually simple, is not eﬃcient and would 516 not ordinarily be used in a production system.15 Exercise 4.11: Instead of representing a frame as a pair of lists, we can represent a frame as a list of bindings, where each binding is a name-value pair. Rewrite the environment operations to use this alternative representation. Exercise 4.12: e procedures set-variable-value!, define- variable! and lookup-variable-value can be expressed in terms of more abstract procedures for traversing the en- vironment structure. Deﬁne abstractions that capture the common paerns and redeﬁne the three procedures in terms of these abstractions. Exercise 4.13: Scheme allows us to create new bindings for variables by means of define, but provides no way to get rid of bindings. Implement for the evaluator a special form make-unbound! that removes the binding of a given symbol from the environment in which the make-unbound! expres- sion is evaluated. is problem is not completely speciﬁed. For example, should we remove only the binding in the ﬁrst frame of the environment? Complete the speciﬁcation and justify any choices you make. 15 e drawback of this representation (as well as the variant in Exercise 4.11) is that the evaluator may have to search through many frames in order to ﬁnd the binding for a given variable. (Such an approach is referred to as deep binding.) One way to avoid this ineﬃciency is to make use of a strategy called lexical addressing, which will be discussed in Section 5.5.6. 517 4.1.4 Running the Evaluator as a Program Given the evaluator, we have in our hands a description (expressed in Lisp) of the process by which Lisp expressions are evaluated. One ad- vantage of expressing the evaluator as a program is that we can run the program. is gives us, running within Lisp, a working model of how Lisp itself evaluates expressions. is can serve as a framework for ex- perimenting with evaluation rules, as we shall do later in this chapter. Our evaluator program reduces expressions ultimately to the appli- cation of primitive procedures. erefore, all that we need to run the evaluator is to create a mechanism that calls on the underlying Lisp system to model the application of primitive procedures. ere must be a binding for each primitive procedure name, so that when eval evaluates the operator of an application of a primitive, it will ﬁnd an object to pass to apply. We thus set up a global environment that associates unique objects with the names of the primitive procedures that can appear in the expressions we will be evaluating. e global environment also includes bindings for the symbols true and false, so that they can be used as variables in expressions to be evaluated. (define (setup-environment) (let ((initial-env (extend-environment (primitive-procedure-names) (primitive-procedure-objects) the-empty-environment))) (define-variable! 'true true initial-env) (define-variable! 'false false initial-env) initial-env)) (define the-global-environment (setup-environment)) It does not maer how we represent the primitive procedure objects, so long as apply can identify and apply them by using the procedures 518 primitive-procedure? and apply-primitive-procedure. We have cho- sen to represent a primitive procedure as a list beginning with the sym- bol primitive and containing a procedure in the underlying Lisp that implements that primitive. (define (primitive-procedure? proc) (tagged-list? proc 'primitive)) (define (primitive-implementation proc) (cadr proc)) setup-environment will get the primitive names and implementation procedures from a list:16 (define primitive-procedures (list (list 'car car) (list 'cdr cdr) (list 'cons cons) (list 'null? null?) ⟨more primitives⟩ )) (define (primitive-procedure-names) (map car primitive-procedures)) (define (primitive-procedure-objects) (map (lambda (proc) (list 'primitive (cadr proc))) primitive-procedures)) To apply a primitive procedure, we simply apply the implementation procedure to the arguments, using the underlying Lisp system:17 16 Any procedure deﬁned in the underlying Lisp can be used as a primitive for the metacircular evaluator. e name of a primitive installed in the evaluator need not be the same as the name of its implementation in the underlying Lisp; the names are the same here because the metacircular evaluator implements Scheme itself. us, for example, we could put (list 'first car) or (list 'square (lambda (x) (* x x))) in the list of primitive-procedures. 17 apply-in-underlying-scheme is the apply procedure we have used in earlier chapters. e metacircular evaluator’s apply procedure (Section 4.1.1) models the 519 (define (apply-primitive-procedure proc args) (apply-in-underlying-scheme (primitive-implementation proc) args)) For convenience in running the metacircular evaluator, we provide a driver loop that models the read-eval-print loop of the underlying Lisp system. It prints a prompt, reads an input expression, evaluates this ex- pression in the global environment, and prints the result. We precede each printed result by an output prompt so as to distinguish the value of the expression from other output that may be printed.18 (define input-prompt ";;; M-Eval input:") (define output-prompt ";;; M-Eval value:") (define (driver-loop) (prompt-for-input input-prompt) (let ((input (read))) (let ((output (eval input the-global-environment))) (announce-output output-prompt) (user-print output))) (driver-loop)) working of this primitive. Having two diﬀerent things called apply leads to a tech- nical problem in running the metacircular evaluator, because deﬁning the metacircular evaluator’s apply will mask the deﬁnition of the primitive. One way around this is to rename the metacircular apply to avoid conﬂict with the name of the primitive proce- dure. We have assumed instead that we have saved a reference to the underlying apply by doing (define apply-in-underlying-scheme apply) before deﬁning the metacircular apply. is allows us to access the original version of apply under a diﬀerent name. 18 e primitive procedure read waits for input from the user, and returns the next complete expression that is typed. For example, if the user types (+ 23 x), read returns a three-element list containing the symbol +, the number 23, and the symbol x. If the user types 'x, read returns a two-element list containing the symbol quote and the symbol x. 520 (define (prompt-for-input string) (newline) (newline) (display string) (newline)) (define (announce-output string) (newline) (display string) (newline)) We use a special printing procedure, user-print, to avoid printing the environment part of a compound procedure, which may be a very long list (or may even contain cycles). (define (user-print object) (if (compound-procedure? object) (display (list 'compound-procedure (procedure-parameters object) (procedure-body object) '<procedure-env>)) (display object))) Now all we need to do to run the evaluator is to initialize the global environment and start the driver loop. Here is a sample interaction: (define the-global-environment (setup-environment)) (driver-loop) ;;; M-Eval input: (define (append x y) (if (null? x) y (cons (car x) (append (cdr x) y)))) ;;; M-Eval value: ok ;;; M-Eval input: (append '(a b c) '(d e f)) ;;; M-Eval value: (a b c d e f) 521 Exercise 4.14: Eva Lu Ator and Louis Reasoner are each experimenting with the metacircular evaluator. Eva types in the deﬁnition of map, and runs some test programs that use it. ey work ﬁne. Louis, in contrast, has installed the system version of map as a primitive for the metacircular evaluator. When he tries it, things go terribly wrong. Ex- plain why Louis’s map fails even though Eva’s works. 4.1.5 Data as Programs In thinking about a Lisp program that evaluates Lisp expressions, an analogy might be helpful. One operational view of the meaning of a program is that a program is a description of an abstract (perhaps in- ﬁnitely large) machine. For example, consider the familiar program to compute factorials: (define (factorial n) (if (= n 1) 1 (* (factorial (- n 1)) n))) We may regard this program as the description of a machine contain- ing parts that decrement, multiply, and test for equality, together with a two-position switch and another factorial machine. (e factorial ma- chine is inﬁnite because it contains another factorial machine within it.) Figure 4.2 is a ﬂow diagram for the factorial machine, showing how the parts are wired together. In a similar way, we can regard the evaluator as a very special ma- chine that takes as input a description of a machine. Given this input, the evaluator conﬁgures itself to emulate the machine described. For ex- ample, if we feed our evaluator the deﬁnition of factorial, as shown in Figure 4.3, the evaluator will be able to compute factorials. 522 factorial 1 1 6 = 720 * -- factorial 1 Figure 4.2: e factorial program, viewed as an abstract machine. From this perspective, our evaluator is seen to be a universal ma- chine. It mimics other machines when these are described as Lisp pro- grams.19 is is striking. Try to imagine an analogous evaluator for 19 e fact that the machines are described in Lisp is inessential. If we give our eval- uator a Lisp program that behaves as an evaluator for some other language, say C, the Lisp evaluator will emulate the C evaluator, which in turn can emulate any ma- chine described as a C program. Similarly, writing a Lisp evaluator in C produces a C program that can execute any Lisp program. e deep idea here is that any evaluator can emulate any other. us, the notion of “what can in principle be computed” (ig- noring practicalities of time and memory required) is independent of the language or the computer, and instead reﬂects an underlying notion of computability. is was ﬁrst demonstrated in a clear way by Alan M. Turing (1912-1954), whose 1936 paper laid the foundations for theoretical computer science. In the paper, Turing presented a simple computational model—now known as a Turing machine—and argued that any “eﬀective process” can be formulated as a program for such a machine. (is argument is known 523 6 eval 720 (define (factorial n) (if (= n 1) 1 (* (factorial (- n 1)) n))) Figure 4.3: e evaluator emulating a factorial machine. electrical circuits. is would be a circuit that takes as input a signal encoding the plans for some other circuit, such as a ﬁlter. Given this in- put, the circuit evaluator would then behave like a ﬁlter with the same description. Such a universal electrical circuit is almost unimaginably complex. It is remarkable that the program evaluator is a rather simple program.20 as the Church-Turing thesis.) Turing then implemented a universal machine, i.e., a Tur- ing machine that behaves as an evaluator for Turing-machine programs. He used this framework to demonstrate that there are well-posed problems that cannot be computed by Turing machines (see Exercise 4.15), and so by implication cannot be formulated as “eﬀective processes.” Turing went on to make fundamental contributions to practical computer science as well. For example, he invented the idea of structuring programs using general-purpose subroutines. See Hodges 1983 for a biography of Turing. 20 Some people ﬁnd it counterintuitive that an evaluator, which is implemented by a relatively simple procedure, can emulate programs that are more complex than the evaluator itself. e existence of a universal evaluator machine is a deep and wonderful property of computation. Recursion theory, a branch of mathematical logic, is concerned with logical limits of computation. Douglas Hofstadter’s beautiful book Gödel, Escher, Bach explores some of these ideas (Hofstadter 1979). 524 Another striking aspect of the evaluator is that it acts as a bridge between the data objects that are manipulated by our programming lan- guage and the programming language itself. Imagine that the evaluator program (implemented in Lisp) is running, and that a user is typing ex- pressions to the evaluator and observing the results. From the perspec- tive of the user, an input expression such as (* x x) is an expression in the programming language, which the evaluator should execute. From the perspective of the evaluator, however, the expression is simply a list (in this case, a list of three symbols: *, x, and x) that is to be manipulated according to a well-deﬁned set of rules. at the user’s programs are the evaluator’s data need not be a source of confusion. In fact, it is sometimes convenient to ignore this distinction, and to give the user the ability to explicitly evaluate a data object as a Lisp expression, by making eval available for use in pro- grams. Many Lisp dialects provide a primitive eval procedure that takes as arguments an expression and an environment and evaluates the ex- pression relative to the environment.21 us, (eval '(* 5 5) user-initial-environment) and (eval (cons '* (list 5 5)) user-initial-environment) will both return 25.22 21 Warning: is eval primitive is not identical to the eval procedure we imple- mented in Section 4.1.1, because it uses actual Scheme environments rather than the sample environment structures we built in Section 4.1.3. ese actual environments cannot be manipulated by the user as ordinary lists; they must be accessed via eval or other special operations. Similarly, the apply primitive we saw earlier is not identical to the metacircular apply, because it uses actual Scheme procedures rather than the procedure objects we constructed in Section 4.1.3 and Section 4.1.4. 22 e implementation of Scheme includes eval, as well as a symbol user- initial-environment that is bound to the initial environment in which the user’s in- 525 Exercise 4.15: Given a one-argument procedure p and an object a, p is said to “halt” on a if evaluating the expres- sion (p a) returns a value (as opposed to terminating with an error message or running forever). Show that it is im- possible to write a procedure halts? that correctly deter- mines whether p halts on a for any procedure p and object a. Use the following reasoning: If you had such a procedure halts?, you could implement the following program: (define (run-forever) (run-forever)) (define (try p) (if (halts? p p) (run-forever) 'halted)) Now consider evaluating the expression (try try) and show that any possible outcome (either halting or running forever) violates the intended behavior of halts?.23 4.1.6 Internal Definitions Our environment model of evaluation and our metacircular evaluator execute deﬁnitions in sequence, extending the environment frame one deﬁnition at a time. is is particularly convenient for interactive pro- gram development, in which the programmer needs to freely mix the application of procedures with the deﬁnition of new procedures. How- ever, if we think carefully about the internal deﬁnitions used to im- plement block structure (introduced in Section 1.1.8), we will ﬁnd that put expressions are evaluated. 23 Although we stipulated that halts? is given a procedure object, notice that this reasoning still applies even if halts? can gain access to the procedure’s text and its environment. is is Turing’s celebrated Halting eorem, which gave the ﬁrst clear example of a non-computable problem, i.e., a well-posed task that cannot be carried out as a computational procedure. 526 name-by-name extension of the environment may not be the best way to deﬁne local variables. Consider a procedure with internal deﬁnitions, such as (define (f x) (define (even? n) (if (= n 0) true (odd? (- n 1)))) (define (odd? n) (if (= n 0) false (even? (- n 1)))) ⟨rest of body of f⟩) Our intention here is that the name odd? in the body of the procedure even? should refer to the procedure odd? that is deﬁned aer even?. e scope of the name odd? is the entire body of f, not just the portion of the body of f starting at the point where the define for odd? occurs. Indeed, when we consider that odd? is itself deﬁned in terms of even?— so that even? and odd? are mutually recursive procedures—we see that the only satisfactory interpretation of the two defines is to regard them as if the names even? and odd? were being added to the environment simultaneously. More generally, in block structure, the scope of a local name is the entire procedure body in which the define is evaluated. As it happens, our interpreter will evaluate calls to f correctly, but for an “accidental” reason: Since the deﬁnitions of the internal proce- dures come ﬁrst, no calls to these procedures will be evaluated until all of them have been deﬁned. Hence, odd? will have been deﬁned by the time even? is executed. In fact, our sequential evaluation mecha- nism will give the same result as a mechanism that directly implements simultaneous deﬁnition for any procedure in which the internal deﬁni- tions come ﬁrst in a body and evaluation of the value expressions for the deﬁned variables doesn’t actually use any of the deﬁned variables. (For an example of a procedure that doesn’t obey these restrictions, so that sequential deﬁnition isn’t equivalent to simultaneous deﬁnition, 527 see Exercise 4.19.)24 ere is, however, a simple way to treat deﬁnitions so that inter- nally deﬁned names have truly simultaneous scope—just create all local variables that will be in the current environment before evaluating any of the value expressions. One way to do this is by a syntax transfor- mation on lambda expressions. Before evaluating the body of a lambda expression, we “scan out” and eliminate all the internal deﬁnitions in the body. e internally deﬁned variables will be created with a let and then set to their values by assignment. For example, the procedure (lambda ⟨vars⟩ (define u ⟨e1⟩) (define v ⟨e2⟩) ⟨e3⟩) would be transformed into (lambda ⟨vars⟩ (let ((u '*unassigned*) (v '*unassigned*)) (set! u ⟨e1⟩) (set! v ⟨e2⟩) ⟨e3⟩)) 24 Wanting programs to not depend on this evaluation mechanism is the reason for the “management is not responsible” remark in Footnote 28 of Chapter 1. By insisting that internal deﬁnitions come ﬁrst and do not use each other while the deﬁnitions are being evaluated, the standard for Scheme leaves implementors some choice in the mechanism used to evaluate these deﬁnitions. e choice of one evaluation rule rather than another here may seem like a small issue, aﬀecting only the interpretation of “badly formed” programs. However, we will see in Section 5.5.6 that moving to a model of simultaneous scoping for internal deﬁnitions avoids some nasty diﬃculties that would otherwise arise in implementing a compiler. 528 where *unassigned* is a special symbol that causes looking up a vari- able to signal an error if an aempt is made to use the value of the not-yet-assigned variable. An alternative strategy for scanning out internal deﬁnitions is shown in Exercise 4.18. Unlike the transformation shown above, this enforces the restriction that the deﬁned variables’ values can be evaluated with- out using any of the variables’ values.25 Exercise 4.16: In this exercise we implement the method just described for interpreting internal deﬁnitions. We as- sume that the evaluator supports let (see Exercise 4.6). a. Change lookup-variable-value (Section 4.1.3) to sig- nal an error if the value it ﬁnds is the symbol *unassigned*. b. Write a procedure scan-out-defines that takes a pro- cedure body and returns an equivalent one that has no internal deﬁnitions, by making the transformation described above. c. Install scan-out-defines in the interpreter, either in make-procedure or in procedure-body (see Section 4.1.3). Which place is beer? Why? Exercise 4.17: Draw diagrams of the environment in eﬀect when evaluating the expression ⟨e3 ⟩ in the procedure in the 25 e standard for Scheme allows for diﬀerent implementation strategies by specifying that it is up to the programmer to obey this restriction, not up to the imple- mentation to enforce it. Some Scheme implementations, including Scheme, use the transformation shown above. us, some programs that don’t obey this restriction will in fact run in such implementations. 529 text, comparing how this will be structured when deﬁni- tions are interpreted sequentially with how it will be struc- tured if deﬁnitions are scanned out as described. Why is there an extra frame in the transformed program? Explain why this diﬀerence in environment structure can never make a diﬀerence in the behavior of a correct program. Design a way to make the interpreter implement the “simultaneous” scope rule for internal deﬁnitions without constructing the extra frame. Exercise 4.18: Consider an alternative strategy for scan- ning out deﬁnitions that translates the example in the text to (lambda ⟨vars⟩ (let ((u '*unassigned*) (v '*unassigned*)) (let ((a ⟨e1⟩) (b ⟨e2⟩)) (set! u a) (set! v b)) ⟨e3⟩)) Here a and b are meant to represent new variable names, created by the interpreter, that do not appear in the user’s program. Consider the solve procedure from Section 3.5.4: (define (solve f y0 dt) (define y (integral (delay dy) y0 dt)) (define dy (stream-map f y)) y) Will this procedure work if internal deﬁnitions are scanned out as shown in this exercise? What if they are scanned out as shown in the text? Explain. 530 Exercise 4.19: Ben Bitdiddle, Alyssa P. Hacker, and Eva Lu Ator are arguing about the desired result of evaluating the expression (let ((a 1)) (define (f x) (define b (+ a x)) (define a 5) (+ a b)) (f 10)) Ben asserts that the result should be obtained using the se- quential rule for define: b is deﬁned to be 11, then a is de- ﬁned to be 5, so the result is 16. Alyssa objects that mutual recursion requires the simultaneous scope rule for internal procedure deﬁnitions, and that it is unreasonable to treat procedure names diﬀerently from other names. us, she argues for the mechanism implemented in Exercise 4.16. is would lead to a being unassigned at the time that the value for b is to be computed. Hence, in Alyssa’s view the procedure should produce an error. Eva has a third opinion. She says that if the deﬁnitions of a and b are truly meant to be simultaneous, then the value 5 for a should be used in evaluating b. Hence, in Eva’s view a should be 5, b should be 15, and the result should be 20. Which (if any) of these view- points do you support? Can you devise a way to implement internal deﬁnitions so that they behave as Eva prefers?26 26 e implementors of Scheme support Alyssa on the following grounds: Eva is in principle correct—the deﬁnitions should be regarded as simultaneous. But it seems diﬃcult to implement a general, eﬃcient mechanism that does what Eva requires. In the absence of such a mechanism, it is beer to generate an error in the diﬃcult cases of simultaneous deﬁnitions (Alyssa’s notion) than to produce an incorrect answer (as Ben would have it). 531 Exercise 4.20: Because internal deﬁnitions look sequen- tial but are actually simultaneous, some people prefer to avoid them entirely, and use the special form letrec in- stead. letrec looks like let, so it is not surprising that the variables it binds are bound simultaneously and have the same scope as each other. e sample procedure f above can be wrien without internal deﬁnitions, but with ex- actly the same meaning, as (define (f x) (letrec ((even? (lambda (n) (if (= n 0) true (odd? (- n 1))))) (odd? (lambda (n) (if (= n 0) false (even? (- n 1)))))) ⟨rest of body of f⟩)) letrec expressions, which have the form (letrec ((⟨var1 ⟩ ⟨exp1 ⟩) . . . (⟨varn ⟩ ⟨expn ⟩)) ⟨body⟩) are a variation on let in which the expressions ⟨expk ⟩ that provide the initial values for the variables ⟨var k ⟩ are eval- uated in an environment that includes all the letrec bind- ings. is permits recursion in the bindings, such as the mutual recursion of even? and odd? in the example above, or the evaluation of 10 factorial with (letrec ((fact (lambda (n) (if (= n 1) 1 (* n (fact (- n 1))))))) (fact 10)) 532 a. Implement letrec as a derived expression, by trans- forming a letrec expression into a let expression as shown in the text above or in Exercise 4.18. at is, the letrec variables should be created with a let and then be assigned their values with set!. b. Louis Reasoner is confused by all this fuss about inter- nal deﬁnitions. e way he sees it, if you don’t like to use define inside a procedure, you can just use let. Illustrate what is loose about his reasoning by draw- ing an environment diagram that shows the environ- ment in which the ⟨rest of body of f ⟩ is evaluated dur- ing evaluation of the expression (f 5), with f deﬁned as in this exercise. Draw an environment diagram for the same evaluation, but with let in place of letrec in the deﬁnition of f. Exercise 4.21: Amazingly, Louis’s intuition in Exercise 4.20 is correct. It is indeed possible to specify recursive proce- dures without using letrec (or even define), although the method for accomplishing this is much more subtle than Louis imagined. e following expression computes 10 fac- torial by applying a recursive factorial procedure:27 ((lambda (n) ((lambda (fact) (fact fact n)) (lambda (ft k) (if (= k 1) 1 (* k (ft ft (- k 1))))))) 10) 27 is example illustrates a programming trick for formulating recursive procedures without using define. e most general trick of this sort is the Y operator, which can be used to give a “pure λ-calculus” implementation of recursion. (See Stoy 1977 for details on the λ-calculus, and Gabriel 1988 for an exposition of the Y operator in Scheme.) 533 a. Check (by evaluating the expression) that this really does compute factorials. Devise an analogous expres- sion for computing Fibonacci numbers. b. Consider the following procedure, which includes mu- tually recursive internal deﬁnitions: (define (f x) (define (even? n) (if (= n 0) true (odd? (- n 1)))) (define (odd? n) (if (= n 0) false (even? (- n 1)))) (even? x)) Fill in the missing expressions to complete an alterna- tive deﬁnition of f, which uses neither internal deﬁ- nitions nor letrec: (define (f x) ((lambda (even? odd?) (even? even? odd? x)) (lambda (ev? od? n) (if (= n 0) true (od? ⟨??⟩ ⟨??⟩ ⟨??⟩))) (lambda (ev? od? n) (if (= n 0) false (ev? ⟨??⟩ ⟨??⟩ ⟨??⟩))))) 4.1.7 Separating Syntactic Analysis from Execution e evaluator implemented above is simple, but it is very ineﬃcient, because the syntactic analysis of expressions is interleaved with their execution. us if a program is executed many times, its syntax is an- alyzed many times. Consider, for example, evaluating (factorial 4) using the following deﬁnition of factorial: (define (factorial n) (if (= n 1) 1 (* (factorial (- n 1)) n))) 534 Each time factorial is called, the evaluator must determine that the body is an if expression and extract the predicate. Only then can it evaluate the predicate and dispatch on its value. Each time it evaluates the expression (* (factorial (- n 1)) n), or the subexpressions (factorial (- n 1)) and (- n 1), the evaluator must perform the case analysis in eval to determine that the expression is an application, and must extract its operator and operands. is analysis is expensive. Performing it repeatedly is wasteful. We can transform the evaluator to be signiﬁcantly more eﬃcient by arranging things so that syntactic analysis is performed only once.28 We split eval, which takes an expression and an environment, into two parts. e procedure analyze takes only the expression. It performs the syntactic analysis and returns a new procedure, the execution procedure, that encapsulates the work to be done in executing the analyzed expres- sion. e execution procedure takes an environment as its argument and completes the evaluation. is saves work because analyze will be called only once on an expression, while the execution procedure may be called many times. With the separation into analysis and execution, eval now becomes (define (eval exp env) ((analyze exp) env)) e result of calling analyze is the execution procedure to be applied to the environment. e analyze procedure is the same case analysis as performed by the original eval of Section 4.1.1, except that the proce- dures to which we dispatch perform only analysis, not full evaluation: (define (analyze exp) 28 is technique is an integral part of the compilation process, which we shall discuss in Chapter 5. Jonathan Rees wrote a Scheme interpreter like this in about 1982 for the T project (Rees and Adams 1982). Marc Feeley (1986) (see also Feeley and Lapalme 1987) independently invented this technique in his master’s thesis. 535 (cond ((self-evaluating? exp) (analyze-self-evaluating exp)) ((quoted? exp) (analyze-quoted exp)) ((variable? exp) (analyze-variable exp)) ((assignment? exp) (analyze-assignment exp)) ((definition? exp) (analyze-definition exp)) ((if? exp) (analyze-if exp)) ((lambda? exp) (analyze-lambda exp)) ((begin? exp) (analyze-sequence (begin-actions exp))) ((cond? exp) (analyze (cond->if exp))) ((application? exp) (analyze-application exp)) (else (error "Unknown expression type: ANALYZE" exp)))) Here is the simplest syntactic analysis procedure, which handles self- evaluating expressions. It returns an execution procedure that ignores its environment argument and just returns the expression: (define (analyze-self-evaluating exp) (lambda (env) exp)) For a quoted expression, we can gain a lile eﬃciency by extracting the text of the quotation only once, in the analysis phase, rather than in the execution phase. (define (analyze-quoted exp) (let ((qval (text-of-quotation exp))) (lambda (env) qval))) Looking up a variable value must still be done in the execution phase, since this depends upon knowing the environment.29 (define (analyze-variable exp) (lambda (env) (lookup-variable-value exp env))) 29 ere is, however, an important part of the variable search that can be done as part of the syntactic analysis. As we will show in Section 5.5.6, one can determine the position in the environment structure where the value of the variable will be found, thus obviating the need to scan the environment for the entry that matches the variable. 536 analyze-assignment also must defer actually seing the variable un- til the execution, when the environment has been supplied. However, the fact that the assignment-value expression can be analyzed (re- cursively) during analysis is a major gain in eﬃciency, because the assignment-value expression will now be analyzed only once. e same holds true for deﬁnitions. (define (analyze-assignment exp) (let ((var (assignment-variable exp)) (vproc (analyze (assignment-value exp)))) (lambda (env) (set-variable-value! var (vproc env) env) 'ok))) (define (analyze-definition exp) (let ((var (definition-variable exp)) (vproc (analyze (definition-value exp)))) (lambda (env) (define-variable! var (vproc env) env) 'ok))) For if expressions, we extract and analyze the predicate, consequent, and alternative at analysis time. (define (analyze-if exp) (let ((pproc (analyze (if-predicate exp))) (cproc (analyze (if-consequent exp))) (aproc (analyze (if-alternative exp)))) (lambda (env) (if (true? (pproc env)) (cproc env) (aproc env))))) Analyzing a lambda expression also achieves a major gain in eﬃciency: We analyze the lambda body only once, even though procedures result- ing from evaluation of the lambda may be applied many times. 537 (define (analyze-lambda exp) (let ((vars (lambda-parameters exp)) (bproc (analyze-sequence (lambda-body exp)))) (lambda (env) (make-procedure vars bproc env)))) Analysis of a sequence of expressions (as in a begin or the body of a lambda expression) is more involved.30 Each expression in the sequence is analyzed, yielding an execution procedure. ese execution proce- dures are combined to produce an execution procedure that takes an environment as argument and sequentially calls each individual execu- tion procedure with the environment as argument. (define (analyze-sequence exps) (define (sequentially proc1 proc2) (lambda (env) (proc1 env) (proc2 env))) (define (loop first-proc rest-procs) (if (null? rest-procs) first-proc (loop (sequentially first-proc (car rest-procs)) (cdr rest-procs)))) (let ((procs (map analyze exps))) (if (null? procs) (error "Empty sequence: ANALYZE")) (loop (car procs) (cdr procs)))) To analyze an application, we analyze the operator and operands and construct an execution procedure that calls the operator execution pro- cedure (to obtain the actual procedure to be applied) and the operand execution procedures (to obtain the actual arguments). We then pass these to execute-application, which is the analog of apply in Section 4.1.1. execute-application diﬀers from apply in that the procedure body for a compound procedure has already been analyzed, so there is 30 See Exercise 4.23 for some insight into the processing of sequences. 538 no need to do further analysis. Instead, we just call the execution pro- cedure for the body on the extended environment. (define (analyze-application exp) (let ((fproc (analyze (operator exp))) (aprocs (map analyze (operands exp)))) (lambda (env) (execute-application (fproc env) (map (lambda (aproc) (aproc env)) aprocs))))) (define (execute-application proc args) (cond ((primitive-procedure? proc) (apply-primitive-procedure proc args)) ((compound-procedure? proc) ((procedure-body proc) (extend-environment (procedure-parameters proc) args (procedure-environment proc)))) (else (error "Unknown procedure type: EXECUTE-APPLICATION" proc)))) Our new evaluator uses the same data structures, syntax procedures, and run-time support procedures as in sections Section 4.1.2, Section 4.1.3, and Section 4.1.4. Exercise 4.22: Extend the evaluator in this section to sup- port the special form let. (See Exercise 4.6.) Exercise 4.23: Alyssa P. Hacker doesn’t understand why analyze-sequence needs to be so complicated. All the other analysis procedures are straightforward transformations of 539 the corresponding evaluation procedures (or eval clauses) in Section 4.1.1. She expected analyze-sequence to look like this: (define (analyze-sequence exps) (define (execute-sequence procs env) (cond ((null? (cdr procs)) ((car procs) env)) (else ((car procs) env) (execute-sequence (cdr procs) env)))) (let ((procs (map analyze exps))) (if (null? procs) (error "Empty sequence: ANALYZE")) (lambda (env) (execute-sequence procs env)))) Eva Lu Ator explains to Alyssa that the version in the text does more of the work of evaluating a sequence at analysis time. Alyssa’s sequence-execution procedure, rather than having the calls to the individual execution procedures built in, loops through the procedures in order to call them: In eﬀect, although the individual expressions in the sequence have been analyzed, the sequence itself has not been. Compare the two versions of analyze-sequence. For ex- ample, consider the common case (typical of procedure bod- ies) where the sequence has just one expression. What work will the execution procedure produced by Alyssa’s program do? What about the execution procedure produced by the program in the text above? How do the two versions com- pare for a sequence with two expressions? 540 Exercise 4.24: Design and carry out some experiments to compare the speed of the original metacircular evaluator with the version in this section. Use your results to esti- mate the fraction of time that is spent in analysis versus execution for various procedures. 4.2 Variations on a Scheme — Lazy Evaluation Now that we have an evaluator expressed as a Lisp program, we can experiment with alternative choices in language design simply by mod- ifying the evaluator. Indeed, new languages are oen invented by ﬁrst writing an evaluator that embeds the new language within an exist- ing high-level language. For example, if we wish to discuss some aspect of a proposed modiﬁcation to Lisp with another member of the Lisp community, we can supply an evaluator that embodies the change. e recipient can then experiment with the new evaluator and send back comments as further modiﬁcations. Not only does the high-level imple- mentation base make it easier to test and debug the evaluator; in addi- tion, the embedding enables the designer to snarf 31 features from the underlying language, just as our embedded Lisp evaluator uses primi- tives and control structure from the underlying Lisp. Only later (if ever) need the designer go to the trouble of building a complete implemen- tation in a low-level language or in hardware. In this section and the next we explore some variations on Scheme that provide signiﬁcant ad- ditional expressive power. 31 Snarf:“To grab, especially a large document or ﬁle for the purpose of using it ei- ther with or without the owner’s permission.” Snarf down: “To snarf, sometimes with the connotation of absorbing, processing, or understanding.” (ese deﬁnitions were snarfed from Steele et al. 1983. See also Raymond 1993.) 541 4.2.1 Normal Order and Applicative Order In Section 1.1, where we began our discussion of models of evaluation, we noted that Scheme is an applicative-order language, namely, that all the arguments to Scheme procedures are evaluated when the procedure is applied. In contrast, normal-order languages delay evaluation of pro- cedure arguments until the actual argument values are needed. Delay- ing evaluation of procedure arguments until the last possible moment (e.g., until they are required by a primitive operation) is called lazy eval- uation.32 Consider the procedure (define (try a b) (if (= a 0) 1 b)) Evaluating (try 0 (/ 1 0)) generates an error in Scheme. With lazy evaluation, there would be no error. Evaluating the expression would return 1, because the argument (/ 1 0) would never be evaluated. An example that exploits lazy evaluation is the deﬁnition of a pro- cedure unless (define (unless condition usual-value exceptional-value) (if condition exceptional-value usual-value)) that can be used in expressions such as (unless (= b 0) (/ a b) (begin (display "exception: returning 0") 0)) is won’t work in an applicative-order language because both the usual value and the exceptional value will be evaluated before unless is called 32 e diﬀerence between the “lazy” terminology and the “normal-order” terminol- ogy is somewhat fuzzy. Generally, “lazy” refers to the mechanisms of particular eval- uators, while “normal-order” refers to the semantics of languages, independent of any particular evaluation strategy. But this is not a hard-and-fast distinction, and the two terminologies are oen used interchangeably. 542 (compare Exercise 1.6). An advantage of lazy evaluation is that some procedures, such as unless, can do useful computation even if evalu- ation of some of their arguments would produce errors or would not terminate. If the body of a procedure is entered before an argument has been evaluated we say that the procedure is non-strict in that argument. If the argument is evaluated before the body of the procedure is entered we say that the procedure is strict in that argument.33 In a purely applicative- order language, all procedures are strict in each argument. In a purely normal-order language, all compound procedures are non-strict in each argument, and primitive procedures may be either strict or non-strict. ere are also languages (see Exercise 4.31) that give programmers de- tailed control over the strictness of the procedures they deﬁne. A striking example of a procedure that can usefully be made non- strict is cons (or, in general, almost any constructor for data structures). One can do useful computation, combining elements to form data struc- tures and operating on the resulting data structures, even if the values of the elements are not known. It makes perfect sense, for instance, to compute the length of a list without knowing the values of the indi- vidual elements in the list. We will exploit this idea in Section 4.2.3 to implement the streams of Chapter 3 as lists formed of non-strict cons pairs. Exercise 4.25: Suppose that (in ordinary applicative-order Scheme) we deﬁne unless as shown above and then deﬁne 33 e “strict” versus “non-strict” terminology means essentially the same thing as “applicative-order” versus “normal-order,” except that it refers to individual procedures and arguments rather than to the language as a whole. At a conference on programming languages you might hear someone say, “e normal-order language Hassle has certain strict primitives. Other procedures take their arguments by lazy evaluation.” 543 factorial in terms of unless as (define (factorial n) (unless (= n 1) (* n (factorial (- n 1))) 1)) What happens if we aempt to evaluate (factorial 5)? Will our deﬁnitions work in a normal-order language? Exercise 4.26: Ben Bitdiddle and Alyssa P. Hacker disagree over the importance of lazy evaluation for implementing things such as unless. Ben points out that it’s possible to implement unless in applicative order as a special form. Alyssa counters that, if one did that, unless would be merely syntax, not a procedure that could be used in conjunction with higher-order procedures. Fill in the details on both sides of the argument. Show how to implement unless as a derived expression (like cond or let), and give an exam- ple of a situation where it might be useful to have unless available as a procedure, rather than as a special form. 4.2.2 An Interpreter with Lazy Evaluation In this section we will implement a normal-order language that is the same as Scheme except that compound procedures are non-strict in each argument. Primitive procedures will still be strict. It is not diﬃcult to modify the evaluator of Section 4.1.1 so that the language it interprets behaves this way. Almost all the required changes center around pro- cedure application. e basic idea is that, when applying a procedure, the interpreter must determine which arguments are to be evaluated and which are to 544 be delayed. e delayed arguments are not evaluated; instead, they are transformed into objects called thunks.34 e thunk must contain the information required to produce the value of the argument when it is needed, as if it had been evaluated at the time of the application. us, the thunk must contain the argument expression and the environment in which the procedure application is being evaluated. e process of evaluating the expression in a thunk is called forc- 35 ing. In general, a thunk will be forced only when its value is needed: when it is passed to a primitive procedure that will use the value of the thunk; when it is the value of a predicate of a conditional; and when it is the value of an operator that is about to be applied as a procedure. One design choice we have available is whether or not to memoize thunks, as we did with delayed objects in Section 3.5.1. With memoization, the ﬁrst time a thunk is forced, it stores the value that is computed. Subsequent forcings simply return the stored value without repeating the computa- tion. We’ll make our interpreter memoize, because this is more eﬃcient for many applications. ere are tricky considerations here, however.36 34 e word thunk was invented by an informal working group that was discussing the implementation of call-by-name in Algol 60. ey observed that most of the analysis of (“thinking about”) the expression could be done at compile time; thus, at run time, the expression would already have been “thunk” about (Ingerman et al. 1960). 35 is is analogous to the use of force on the delayed objects that were introduced in Chapter 3 to represent streams. e critical diﬀerence between what we are doing here and what we did in Chapter 3 is that we are building delaying and forcing into the evaluator, and thus making this uniform and automatic throughout the language. 36 Lazy evaluation combined with memoization is sometimes referred to as call-by- need argument passing, in contrast to call-by-name argument passing. (Call-by-name, introduced in Algol 60, is similar to non-memoized lazy evaluation.) As language de- signers, we can build our evaluator to memoize, not to memoize, or leave this an option for programmers (Exercise 4.31). As you might expect from Chapter 3, these choices raise issues that become both subtle and confusing in the presence of assignments. (See Exercise 4.27 and Exercise 4.29.) An excellent article by Clinger (1982) aempts to clar- 545 Modifying the evaluator e main diﬀerence between the lazy evaluator and the one in Section 4.1 is in the handling of procedure applications in eval and apply. e application? clause of eval becomes ((application? exp) (apply (actual-value (operator exp) env) (operands exp) env)) is is almost the same as the application? clause of eval in Sec- tion 4.1.1. For lazy evaluation, however, we call apply with the operand expressions, rather than the arguments produced by evaluating them. Since we will need the environment to construct thunks if the argu- ments are to be delayed, we must pass this as well. We still evaluate the operator, because apply needs the actual procedure to be applied in order to dispatch on its type (primitive versus compound) and apply it. Whenever we need the actual value of an expression, we use (define (actual-value exp env) (force-it (eval exp env))) instead of just eval, so that if the expression’s value is a thunk, it will be forced. Our new version of apply is also almost the same as the version in Section 4.1.1. e diﬀerence is that eval has passed in unevaluated operand expressions: For primitive procedures (which are strict), we evaluate all the arguments before applying the primitive; for compound procedures (which are non-strict) we delay all the arguments before ap- plying the procedure. ify the multiple dimensions of confusion that arise here. 546 (define (apply procedure arguments env) (cond ((primitive-procedure? procedure) (apply-primitive-procedure procedure (list-of-arg-values arguments env))) ; changed ((compound-procedure? procedure) (eval-sequence (procedure-body procedure) (extend-environment (procedure-parameters procedure) (list-of-delayed-args arguments env) ; changed (procedure-environment procedure)))) (else (error "Unknown procedure type: APPLY" procedure)))) e procedures that process the arguments are just like list-of-values from Section 4.1.1, except that list-of-delayed-args delays the argu- ments instead of evaluating them, and list-of-arg-values uses actual- value instead of eval: (define (list-of-arg-values exps env) (if (no-operands? exps) '() (cons (actual-value (first-operand exps) env) (list-of-arg-values (rest-operands exps) env)))) (define (list-of-delayed-args exps env) (if (no-operands? exps) '() (cons (delay-it (first-operand exps) env) (list-of-delayed-args (rest-operands exps) env)))) 547 e other place we must change the evaluator is in the handling of if, where we must use actual-value instead of eval to get the value of the predicate expression before testing whether it is true or false: (define (eval-if exp env) (if (true? (actual-value (if-predicate exp) env)) (eval (if-consequent exp) env) (eval (if-alternative exp) env))) Finally, we must change the driver-loop procedure (Section 4.1.4) to use actual-value instead of eval, so that if a delayed value is prop- agated back to the read-eval-print loop, it will be forced before being printed. We also change the prompts to indicate that this is the lazy evaluator: (define input-prompt ";;; L-Eval input:") (define output-prompt ";;; L-Eval value:") (define (driver-loop) (prompt-for-input input-prompt) (let ((input (read))) (let ((output (actual-value input the-global-environment))) (announce-output output-prompt) (user-print output))) (driver-loop)) With these changes made, we can start the evaluator and test it. e successful evaluation of the try expression discussed in Section 4.2.1 indicates that the interpreter is performing lazy evaluation: (define the-global-environment (setup-environment)) (driver-loop) ;;; L-Eval input: (define (try a b) (if (= a 0) 1 b)) 548 ;;; L-Eval value: ok ;;; L-Eval input: (try 0 (/ 1 0)) ;;; L-Eval value: 1 Representing thunks Our evaluator must arrange to create thunks when procedures are ap- plied to arguments and to force these thunks later. A thunk must pack- age an expression together with the environment, so that the argument can be produced later. To force the thunk, we simply extract the ex- pression and environment from the thunk and evaluate the expression in the environment. We use actual-value rather than eval so that in case the value of the expression is itself a thunk, we will force that, and so on, until we reach something that is not a thunk: (define (force-it obj) (if (thunk? obj) (actual-value (thunk-exp obj) (thunk-env obj)) obj)) One easy way to package an expression with an environment is to make a list containing the expression and the environment. us, we create a thunk as follows: (define (delay-it exp env) (list 'thunk exp env)) (define (thunk? obj) (tagged-list? obj 'thunk)) (define (thunk-exp thunk) (cadr thunk)) (define (thunk-env thunk) (caddr thunk)) 549 Actually, what we want for our interpreter is not quite this, but rather thunks that have been memoized. When a thunk is forced, we will turn it into an evaluated thunk by replacing the stored expression with its value and changing the thunk tag so that it can be recognized as already evaluated.37 (define (evaluated-thunk? obj) (tagged-list? obj 'evaluated-thunk)) (define (thunk-value evaluated-thunk) (cadr evaluated-thunk)) (define (force-it obj) (cond ((thunk? obj) (let ((result (actual-value (thunk-exp obj) (thunk-env obj)))) (set-car! obj 'evaluated-thunk) (set-car! (cdr obj) result) ; replace exp with its value (set-cdr! (cdr obj) '()) ; forget unneeded env result)) ((evaluated-thunk? obj) (thunk-value obj)) (else obj))) Notice that the same delay-it procedure works both with and without memoization. 37 Notice that we also erase the env from the thunk once the expression’s value has been computed. is makes no diﬀerence in the values returned by the interpreter. It does help save space, however, because removing the reference from the thunk to the env once it is no longer needed allows this structure to be garbage-collected and its space recycled, as we will discuss in Section 5.3. Similarly, we could have allowed unneeded environments in the memoized delayed objects of Section 3.5.1 to be garbage-collected, by having memo-proc do something like (set! proc '()) to discard the procedure proc (which includes the environment in which the delay was evaluated) aer storing its value. 550 Exercise 4.27: Suppose we type in the following deﬁnitions to the lazy evaluator: (define count 0) (define (id x) (set! count (+ count 1)) x) Give the missing values in the following sequence of inter- actions, and explain your answers.38 (define w (id (id 10))) ;;; L-Eval input: count ;;; L-Eval value: ⟨response⟩ ;;; L-Eval input: w ;;; L-Eval value: ⟨response⟩ ;;; L-Eval input: count ;;; L-Eval value: ⟨response⟩ Exercise 4.28: eval uses actual-value rather than eval to evaluate the operator before passing it to apply, in or- der to force the value of the operator. Give an example that demonstrates the need for this forcing. Exercise 4.29: Exhibit a program that you would expect to run much more slowly without memoization than with 38 is exercise demonstrates that the interaction between lazy evaluation and side eﬀects can be very confusing. is is just what you might expect from the discussion in Chapter 3. 551 memoization. Also, consider the following interaction, where the id procedure is deﬁned as in Exercise 4.27 and count starts at 0: (define (square x) (* x x)) ;;; L-Eval input: (square (id 10)) ;;; L-Eval value: ⟨response⟩ ;;; L-Eval input: count ;;; L-Eval value: ⟨response⟩ Give the responses both when the evaluator memoizes and when it does not. Exercise 4.30: Cy D. Fect, a reformed C programmer, is worried that some side eﬀects may never take place, be- cause the lazy evaluator doesn’t force the expressions in a sequence. Since the value of an expression in a sequence other than the last one is not used (the expression is there only for its eﬀect, such as assigning to a variable or print- ing), there can be no subsequent use of this value (e.g., as an argument to a primitive procedure) that will cause it to be forced. Cy thus thinks that when evaluating sequences, we must force all expressions in the sequence except the ﬁnal one. He proposes to modify eval-sequence from Section 4.1.1 to use actual-value rather than eval: (define (eval-sequence exps env) (cond ((last-exp? exps) (eval (first-exp exps) env)) (else (actual-value (first-exp exps) env) 552 (eval-sequence (rest-exps exps) env)))) a. Ben Bitdiddle thinks Cy is wrong. He shows Cy the for-each procedure described in Exercise 2.23, which gives an important example of a sequence with side eﬀects: (define (for-each proc items) (if (null? items) 'done (begin (proc (car items)) (for-each proc (cdr items))))) He claims that the evaluator in the text (with the orig- inal eval-sequence) handles this correctly: ;;; L-Eval input: (for-each (lambda (x) (newline) (display x)) (list 57 321 88)) 57 321 88 ;;; L-Eval value: done Explain why Ben is right about the behavior of for- each. b. Cy agrees that Ben is right about the for-each exam- ple, but says that that’s not the kind of program he was thinking about when he proposed his change to eval-sequence. He deﬁnes the following two proce- dures in the lazy evaluator: 553 (define (p1 x) (set! x (cons x '(2))) x) (define (p2 x) (define (p e) e x) (p (set! x (cons x '(2))))) What are the values of (p1 1) and (p2 1) with the original eval-sequence? What would the values be with Cy’s proposed change to eval-sequence? c. Cy also points out that changing eval-sequence as he proposes does not aﬀect the behavior of the example in part a. Explain why this is true. d. How do you think sequences ought to be treated in the lazy evaluator? Do you like Cy’s approach, the ap- proach in the text, or some other approach? Exercise 4.31: e approach taken in this section is some- what unpleasant, because it makes an incompatible change to Scheme. It might be nicer to implement lazy evaluation as an upward-compatible extension, that is, so that ordinary Scheme programs will work as before. We can do this by extending the syntax of procedure declarations to let the user control whether or not arguments are to be delayed. While we’re at it, we may as well also give the user the choice between delaying with and without memoization. For example, the deﬁnition (define (f a (b lazy) c (d lazy-memo)) . . .) 554 would deﬁne f to be a procedure of four arguments, where the ﬁrst and third arguments are evaluated when the pro- cedure is called, the second argument is delayed, and the fourth argument is both delayed and memoized. us, or- dinary procedure deﬁnitions will produce the same behav- ior as ordinary Scheme, while adding the lazy-memo dec- laration to each parameter of every compound procedure will produce the behavior of the lazy evaluator deﬁned in this section. Design and implement the changes required to produce such an extension to Scheme. You will have to implement new syntax procedures to handle the new syn- tax for define. You must also arrange for eval or apply to determine when arguments are to be delayed, and to force or delay arguments accordingly, and you must arrange for forcing to memoize or not, as appropriate. 4.2.3 Streams as Lazy Lists In Section 3.5.1, we showed how to implement streams as delayed lists. We introduced special forms delay and cons-stream, which allowed us to construct a “promise” to compute the cdr of a stream, without actually fulﬁlling that promise until later. We could use this general technique of introducing special forms whenever we need more control over the evaluation process, but this is awkward. For one thing, a spe- cial form is not a ﬁrst-class object like a procedure, so we cannot use it together with higher-order procedures.39 Additionally, we were forced to create streams as a new kind of data object similar but not identical to lists, and this required us to reimplement many ordinary list operations 39 is is precisely the issue with the unless procedure, as in Exercise 4.26. 555 (map, append, and so on) for use with streams. With lazy evaluation, streams and lists can be identical, so there is no need for special forms or for separate list and stream operations. All we need to do is to arrange maers so that cons is non-strict. One way to accomplish this is to extend the lazy evaluator to allow for non-strict primitives, and to implement cons as one of these. An easier way is to re- call (Section 2.1.3) that there is no fundamental need to implement cons as a primitive at all. Instead, we can represent pairs as procedures:40 (define (cons x y) (lambda (m) (m x y))) (define (car z) (z (lambda (p q) p))) (define (cdr z) (z (lambda (p q) q))) In terms of these basic operations, the standard deﬁnitions of the list operations will work with inﬁnite lists (streams) as well as ﬁnite ones, and the stream operations can be implemented as list operations. Here are some examples: (define (list-ref items n) (if (= n 0) (car items) (list-ref (cdr items) (- n 1)))) (define (map proc items) (if (null? items) '() (cons (proc (car items)) (map proc (cdr items))))) (define (scale-list items factor) (map (lambda (x) (* x factor)) items)) 40 is is the procedural representation described in Exercise 2.4. Essentially any pro- cedural representation (e.g., a message-passing implementation) would do as well. No- tice that we can install these deﬁnitions in the lazy evaluator simply by typing them at the driver loop. If we had originally included cons, car, and cdr as primitives in the global environment, they will be redeﬁned. (Also see Exercise 4.33 and Exercise 4.34.) 556 (define (add-lists list1 list2) (cond ((null? list1) list2) ((null? list2) list1) (else (cons (+ (car list1) (car list2)) (add-lists (cdr list1) (cdr list2)))))) (define ones (cons 1 ones)) (define integers (cons 1 (add-lists ones integers))) ;;; L-Eval input: (list-ref integers 17) ;;; L-Eval value: 18 Note that these lazy lists are even lazier than the streams of Chapter 3: e car of the list, as well as the cdr, is delayed.41 In fact, even accessing the car or cdr of a lazy pair need not force the value of a list element. e value will be forced only when it is really needed—e.g., for use as the argument of a primitive, or to be printed as an answer. Lazy pairs also help with the problem that arose with streams in Sec- tion 3.5.4, where we found that formulating stream models of systems with loops may require us to sprinkle our programs with explicit delay operations, beyond the ones supplied by cons-stream. With lazy evalu- ation, all arguments to procedures are delayed uniformly. For instance, we can implement procedures to integrate lists and solve diﬀerential equations as we originally intended in Section 3.5.4: (define (integral integrand initial-value dt) (define int (cons initial-value (add-lists (scale-list integrand dt) int))) int) 41 is permits us to create delayed versions of more general kinds of list structures, not just sequences. Hughes 1990 discusses some applications of “lazy trees.” 557 (define (solve f y0 dt) (define y (integral dy y0 dt)) (define dy (map f y)) y) ;;; L-Eval input: (list-ref (solve (lambda (x) x) 1 0.001) 1000) ;;; L-Eval value: 2.716924 Exercise 4.32: Give some examples that illustrate the dif- ference between the streams of Chapter 3 and the “lazier” lazy lists described in this section. How can you take ad- vantage of this extra laziness? Exercise 4.33: Ben Bitdiddle tests the lazy list implemen- tation given above by evaluating the expression: (car '(a b c)) To his surprise, this produces an error. Aer some thought, he realizes that the “lists” obtained by reading in quoted expressions are diﬀerent from the lists manipulated by the new deﬁnitions of cons, car, and cdr. Modify the evalua- tor’s treatment of quoted expressions so that quoted lists typed at the driver loop will produce true lazy lists. Exercise 4.34: Modify the driver loop for the evaluator so that lazy pairs and lists will print in some reasonable way. (What are you going to do about inﬁnite lists?) You may also need to modify the representation of lazy pairs so that the evaluator can identify them in order to print them. 558 4.3 Variations on a Scheme — Nondeterministic Computing In this section, we extend the Scheme evaluator to support a program- ming paradigm called nondeterministic computing by building into the evaluator a facility to support automatic search. is is a much more profound change to the language than the introduction of lazy evalua- tion in Section 4.2. Nondeterministic computing, like stream processing, is useful for “generate and test” applications. Consider the task of starting with two lists of positive integers and ﬁnding a pair of integers—one from the ﬁrst list and one from the second list—whose sum is prime. We saw how to handle this with ﬁnite sequence operations in Section 2.2.3 and with inﬁnite streams in Section 3.5.3. Our approach was to generate the se- quence of all possible pairs and ﬁlter these to select the pairs whose sum is prime. Whether we actually generate the entire sequence of pairs ﬁrst as in Chapter 2, or interleave the generating and ﬁltering as in Chap- ter 3, is immaterial to the essential image of how the computation is organized. e nondeterministic approach evokes a diﬀerent image. Imagine simply that we choose (in some way) a number from the ﬁrst list and a number from the second list and require (using some mechanism) that their sum be prime. is is expressed by following procedure: (define (prime-sum-pair list1 list2) (let ((a (an-element-of list1)) (b (an-element-of list2))) (require (prime? (+ a b))) (list a b))) It might seem as if this procedure merely restates the problem, rather than specifying a way to solve it. Nevertheless, this is a legitimate non- 559 deterministic program.42 e key idea here is that expressions in a nondeterministic language can have more than one possible value. For instance, an-element-of might return any element of the given list. Our nondeterministic pro- gram evaluator will work by automatically choosing a possible value and keeping track of the choice. If a subsequent requirement is not met, the evaluator will try a diﬀerent choice, and it will keep trying new choices until the evaluation succeeds, or until we run out of choices. Just as the lazy evaluator freed the programmer from the details of how values are delayed and forced, the nondeterministic program evaluator will free the programmer from the details of how choices are made. It is instructive to contrast the diﬀerent images of time evoked by nondeterministic evaluation and stream processing. Stream processing uses lazy evaluation to decouple the time when the stream of possible answers is assembled from the time when the actual stream elements are produced. e evaluator supports the illusion that all the possible an- swers are laid out before us in a timeless sequence. With nondetermin- istic evaluation, an expression represents the exploration of a set of pos- sible worlds, each determined by a set of choices. Some of the possible worlds lead to dead ends, while others have useful values. e nonde- terministic program evaluator supports the illusion that time branches, and that our programs have diﬀerent possible execution histories. When 42 We assume that we have previously deﬁned a procedure prime? that tests whether numbers are prime. Even with prime? deﬁned, the prime-sum-pair procedure may look suspiciously like the unhelpful “pseudo-Lisp” aempt to deﬁne the square-root function, which we described at the beginning of Section 1.1.7. In fact, a square-root procedure along those lines can actually be formulated as a nondeterministic program. By incorporating a search mechanism into the evaluator, we are eroding the distinc- tion between purely declarative descriptions and imperative speciﬁcations of how to compute answers. We’ll go even farther in this direction in Section 4.4. 560 we reach a dead end, we can revisit a previous choice point and proceed along a diﬀerent branch. e nondeterministic program evaluator implemented below is called the amb evaluator because it is based on a new special form called amb. We can type the above deﬁnition of prime-sum-pair at the amb evalu- ator driver loop (along with deﬁnitions of prime?, an-element-of, and require) and run the procedure as follows: ;;; Amb-Eval input: (prime-sum-pair '(1 3 5 8) '(20 35 110)) ;;; Starting a new problem ;;; Amb-Eval value: (3 20) e value returned was obtained aer the evaluator repeatedly chose elements from each of the lists, until a successful choice was made. Section 4.3.1 introduces amb and explains how it supports nondeter- minism through the evaluator’s automatic search mechanism. Section 4.3.2 presents examples of nondeterministic programs, and Section 4.3.3 gives the details of how to implement the amb evaluator by modifying the ordinary Scheme evaluator. 4.3.1 Amb and Search To extend Scheme to support nondeterminism, we introduce a new spe- cial form called amb.43 e expression (amb ⟨e1 ⟩ ⟨e2 ⟩ . . . ⟨en ⟩) returns the value of one of the n expressions ⟨ei ⟩ “ambiguously.” For example, the expression 43 eidea of amb for nondeterministic programming was ﬁrst described in 1961 by John McCarthy (see McCarthy 1963). 561 (list (amb 1 2 3) (amb 'a 'b)) can have six possible values: (1 a) (1 b) (2 a) (2 b) (3 a) (3 b) amb with a single choice produces an ordinary (single) value. ambwith no choices—the expression (amb)—is an expression with no acceptable values. Operationally, we can think of (amb) as an expres- sion that when evaluated causes the computation to “fail”: e compu- tation aborts and no value is produced. Using this idea, we can express the requirement that a particular predicate expression p must be true as follows: (define (require p) (if (not p) (amb))) With amb and require, we can implement the an-element-of proce- dure used above: (define (an-element-of items) (require (not (null? items))) (amb (car items) (an-element-of (cdr items)))) an-element-of fails if the list is empty. Otherwise it ambiguously re- turns either the ﬁrst element of the list or an element chosen from the rest of the list. We can also express inﬁnite ranges of choices. e following proce- dure potentially returns any integer greater than or equal to some given n: (define (an-integer-starting-from n) (amb n (an-integer-starting-from (+ n 1)))) is is like the stream procedure integers-starting-from described in Section 3.5.2, but with an important diﬀerence: e stream procedure 562 returns an object that represents the sequence of all integers beginning with n, whereas the amb procedure returns a single integer.44 Abstractly, we can imagine that evaluating an amb expression causes time to split into branches, where the computation continues on each branch with one of the possible values of the expression. We say that amb represents a nondeterministic choice point. If we had a machine with a suﬃcient number of processors that could be dynamically allocated, we could implement the search in a straightforward way. Execution would proceed as in a sequential machine, until an amb expression is encoun- tered. At this point, more processors would be allocated and initialized to continue all of the parallel executions implied by the choice. Each processor would proceed sequentially as if it were the only choice, until it either terminates by encountering a failure, or it further subdivides, or it ﬁnishes.45 On the other hand, if we have a machine that can execute only one process (or a few concurrent processes), we must consider the alterna- tives sequentially. One could imagine modifying an evaluator to pick at random a branch to follow whenever it encounters a choice point. 44 In actuality, the distinction between nondeterministically returning a single choice and returning all choices depends somewhat on our point of view. From the perspective of the code that uses the value, the nondeterministic choice returns a single value. From the perspective of the programmer designing the code, the nondeterministic choice potentially returns all possible values, and the computation branches so that each value is investigated separately. 45 One might object that this is a hopelessly ineﬃcient mechanism. It might require millions of processors to solve some easily stated problem this way, and most of the time most of those processors would be idle. is objection should be taken in the context of history. Memory used to be considered just such an expensive commodity. In 1964 a megabyte of cost about $400,000. Now every personal computer has many megabytes of , and most of the time most of that is unused. It is hard to underestimate the cost of mass-produced electronics. 563 Random choice, however, can easily lead to failing values. We might try running the evaluator over and over, making random choices and hoping to ﬁnd a non-failing value, but it is beer to systematically search all possible execution paths. e amb evaluator that we will develop and work with in this section implements a systematic search as follows: When the evaluator encounters an application of amb, it initially selects the ﬁrst alternative. is selection may itself lead to a further choice. e evaluator will always initially choose the ﬁrst alternative at each choice point. If a choice results in a failure, then the evaluator automagically46 backtracks to the most recent choice point and tries the next alternative. If it runs out of alternatives at any choice point, the evaluator will back up to the previous choice point and resume from there. is process leads to a search strategy known as depth-ﬁrst search or chronological backtracking.47 46 Automagically: “Automatically, but in a way which, for some reason (typically be- cause it is too complicated, or too ugly, or perhaps even too trivial), the speaker doesn’t feel like explaining.” (Steele et al. 1983, Raymond 1993) 47 e integration of automatic search strategies into programming languages has had a long and checkered history. e ﬁrst suggestions that nondeterministic algo- rithms might be elegantly encoded in a programming language with search and au- tomatic backtracking came from Robert Floyd (1967). Carl Hewi (1969) invented a programming language called Planner that explicitly supported automatic chronolog- ical backtracking, providing for a built-in depth-ﬁrst search strategy. Sussman et al. (1971) implemented a subset of this language, called MicroPlanner, which was used to support work in problem solving and robot planning. Similar ideas, arising from logic and theorem proving, led to the genesis in Edinburgh and Marseille of the ele- gant language Prolog (which we will discuss in Section 4.4). Aer suﬃcient frustration with automatic search, McDermo and Sussman (1972) developed a language called Conniver, which included mechanisms for placing the search strategy under program- mer control. is proved unwieldy, however, and Sussman and Stallman 1975 found a more tractable approach while investigating methods of symbolic analysis for electrical circuits. ey developed a non-chronological backtracking scheme that was based on 564 Driver loop e driver loop for the amb evaluator has some unusual properties. It reads an expression and prints the value of the ﬁrst non-failing execu- tion, as in the prime-sum-pair example shown above. If we want to see the value of the next successful execution, we can ask the interpreter to backtrack and aempt to generate a second non-failing execution. is is signaled by typing the symbol try-again. If any expression except try-again is given, the interpreter will start a new problem, discarding the unexplored alternatives in the previous problem. Here is a sample interaction: ;;; Amb-Eval input: (prime-sum-pair '(1 3 5 8) '(20 35 110)) ;;; Starting a new problem ;;; Amb-Eval value: (3 20) ;;; Amb-Eval input: try-again ;;; Amb-Eval value: tracing out the logical dependencies connecting facts, a technique that has come to be known as dependency-directed backtracking. Although their method was complex, it pro- duced reasonably eﬃcient programs because it did lile redundant search. Doyle (1979) and McAllester (1978; 1980) generalized and clariﬁed the methods of Stallman and Suss- man, developing a new paradigm for formulating search that is now called truth main- tenance. Modern problem-solving systems all use some form of truth-maintenance sys- tem as a substrate. See Forbus and deKleer 1993 for a discussion of elegant ways to build truth-maintenance systems and applications using truth maintenance. Zabih et al. 1987 describes a nondeterministic extension to Scheme that is based on amb; it is similar to the interpreter described in this section, but more sophisticated, because it uses dependency-directed backtracking rather than chronological backtracking. Win- ston 1992 gives an introduction to both kinds of backtracking. 565 (3 110) ;;; Amb-Eval input: try-again ;;; Amb-Eval value: (8 35) ;;; Amb-Eval input: try-again ;;; There are no more values of (prime-sum-pair (quote (1 3 5 8)) (quote (20 35 110))) ;;; Amb-Eval input: (prime-sum-pair '(19 27 30) '(11 36 58)) ;;; Starting a new problem ;;; Amb-Eval value: (30 11) Exercise 4.35: Write a procedure an-integer-between that returns an integer between two given bounds. is can be used to implement a procedure that ﬁnds Pythagorean triples, i.e., triples of integers (i, j, k) between the given bounds such that i ≤ j and i 2 + j 2 = k 2 , as follows: (define (a-pythagorean-triple-between low high) (let ((i (an-integer-between low high))) (let ((j (an-integer-between i high))) (let ((k (an-integer-between j high))) (require (= (+ (* i i) (* j j)) (* k k))) (list i j k))))) Exercise 4.36: Exercise 3.69 discussed how to generate the stream of all Pythagorean triples, with no upper bound on 566 the size of the integers to be searched. Explain why simply replacing an-integer-between by an-integer-starting- from in the procedure in Exercise 4.35 is not an adequate way to generate arbitrary Pythagorean triples. Write a pro- cedure that actually will accomplish this. (at is, write a procedure for which repeatedly typing try-again would in principle eventually generate all Pythagorean triples.) Exercise 4.37: Ben Bitdiddle claims that the following method for generating Pythagorean triples is more eﬃcient than the one in Exercise 4.35. Is he correct? (Hint: Consider the number of possibilities that must be explored.) (define (a-pythagorean-triple-between low high) (let ((i (an-integer-between low high)) (hsq (* high high))) (let ((j (an-integer-between i high))) (let ((ksq (+ (* i i) (* j j)))) (require (>= hsq ksq)) (let ((k (sqrt ksq))) (require (integer? k)) (list i j k)))))) 4.3.2 Examples of Nondeterministic Programs Section 4.3.3 describes the implementation of the amb evaluator. First, however, we give some examples of how it can be used. e advantage of nondeterministic programming is that we can suppress the details of how search is carried out, thereby expressing our programs at a higher level of abstraction. 567 Logic Puzzles e following puzzle (taken from Dinesman 1968) is typical of a large class of simple logic puzzles: Baker, Cooper, Fletcher, Miller, and Smith live on diﬀer- ent ﬂoors of an apartment house that contains only ﬁve ﬂoors. Baker does not live on the top ﬂoor. Cooper does not live on the boom ﬂoor. Fletcher does not live on ei- ther the top or the boom ﬂoor. Miller lives on a higher ﬂoor than does Cooper. Smith does not live on a ﬂoor adja- cent to Fletcher’s. Fletcher does not live on a ﬂoor adjacent to Cooper’s. Where does everyone live? We can determine who lives on each ﬂoor in a straightforward way by enumerating all the possibilities and imposing the given restrictions:48 (define (multiple-dwelling) (let ((baker (amb 1 2 3 4 5)) (cooper (amb 1 2 3 4 5)) (fletcher (amb 1 2 3 4 5)) (miller (amb 1 2 3 4 5)) (smith (amb 1 2 3 4 5))) (require (distinct? (list baker cooper fletcher miller smith))) (require (not (= baker 5))) 48 Our program uses the following procedure to determine if the elements of a list are distinct: (define (distinct? items) (cond ((null? items) true) ((null? (cdr items)) true) ((member (car items) (cdr items)) false) (else (distinct? (cdr items))))) member is like memq except that it uses equal? instead of eq? to test for equality. 568 (require (not (= cooper 1))) (require (not (= fletcher 5))) (require (not (= fletcher 1))) (require (> miller cooper)) (require (not (= (abs (- smith fletcher)) 1))) (require (not (= (abs (- fletcher cooper)) 1))) (list (list 'baker baker) (list 'cooper cooper) (list 'fletcher fletcher) (list 'miller miller) (list 'smith smith)))) Evaluating the expression (multiple-dwelling) produces the result ((baker 3) (cooper 2) (fletcher 4) (miller 5) (smith 1)) Although this simple procedure works, it is very slow. Exercise 4.39 and Exercise 4.40 discuss some possible improvements. Exercise 4.38: Modify the multiple-dwelling procedure to omit the requirement that Smith and Fletcher do not live on adjacent ﬂoors. How many solutions are there to this modiﬁed puzzle? Exercise 4.39: Does the order of the restrictions in the multiple- dwelling procedure aﬀect the answer? Does it aﬀect the time to ﬁnd an answer? If you think it maers, demonstrate a faster program obtained from the given one by reordering the restrictions. If you think it does not maer, argue your case. Exercise 4.40: In the multiple dwelling problem, how many sets of assignments are there of people to ﬂoors, both be- fore and aer the requirement that ﬂoor assignments be distinct? It is very ineﬃcient to generate all possible assign- ments of people to ﬂoors and then leave it to backtracking 569 to eliminate them. For example, most of the restrictions de- pend on only one or two of the person-ﬂoor variables, and can thus be imposed before ﬂoors have been selected for all the people. Write and demonstrate a much more eﬃ- cient nondeterministic procedure that solves this problem based upon generating only those possibilities that are not already ruled out by previous restrictions. (Hint: is will require a nest of let expressions.) Exercise 4.41: Write an ordinary Scheme program to solve the multiple dwelling puzzle. Exercise 4.42: Solve the following “Liars” puzzle (from Phillips 1934): Five schoolgirls sat for an examination. eir parents—so they thought—showed an undue degree of interest in the result. ey therefore agreed that, in writing home about the examination, each girl should make one true statement and one untrue one. e following are the relevant passages from their leers: • Bey: “Kiy was second in the examination. I was only third.” • Ethel: “You’ll be glad to hear that I was on top. Joan was 2nd.” • Joan: “I was third, and poor old Ethel was boom.” • Kiy: “I came out second. Mary was only fourth.” • Mary: “I was fourth. Top place was taken by Bey.” 570 What in fact was the order in which the ﬁve girls were placed? Exercise 4.43: Use the amb evaluator to solve the following puzzle:49 Mary Ann Moore’s father has a yacht and so has each of his four friends: Colonel Downing, Mr. Hall, Sir Barnacle Hood, and Dr. Parker. Each of the ﬁve also has one daugh- ter and each has named his yacht aer a daughter of one of the others. Sir Barnacle’s yacht is the Gabrielle, Mr. Moore owns the Lorna; Mr. Hall the Rosalind. e Melissa, owned by Colonel Downing, is named aer Sir Barnacle’s daugh- ter. Gabrielle’s father owns the yacht that is named aer Dr. Parker’s daughter. Who is Lorna’s father? Try to write the program so that it runs eﬃciently (see Ex- ercise 4.40). Also determine how many solutions there are if we are not told that Mary Ann’s last name is Moore. Exercise 4.44: Exercise 2.42 described the “eight-queens puzzle” of placing queens on a chessboard so that no two at- tack each other. Write a nondeterministic program to solve this puzzle. Parsing natural language Programs designed to accept natural language as input usually start by aempting to parse the input, that is, to match the input against some grammatical structure. For example, we might try to recognize simple 49 is is taken from a booklet called “Problematical Recreations,” published in the 1960s by Lion Industries, where it is aributed to the Kansas State Engineer. 571 sentences consisting of an article followed by a noun followed by a verb, such as “e cat eats.” To accomplish such an analysis, we must be able to identify the parts of speech of individual words. We could start with some lists that classify various words:50 (define nouns '(noun student professor cat class)) (define verbs '(verb studies lectures eats sleeps)) (define articles '(article the a)) We also need a grammar, that is, a set of rules describing how gram- matical elements are composed from simpler elements. A very simple grammar might stipulate that a sentence always consists of two pieces— a noun phrase followed by a verb—and that a noun phrase consists of an article followed by a noun. With this grammar, the sentence “e cat eats” is parsed as follows: (sentence (noun-phrase (article the) (noun cat)) (verb eats)) We can generate such a parse with a simple program that has separate procedures for each of the grammatical rules. To parse a sentence, we identify its two constituent pieces and return a list of these two ele- ments, tagged with the symbol sentence: (define (parse-sentence) (list 'sentence (parse-noun-phrase) (parse-word verbs))) A noun phrase, similarly, is parsed by ﬁnding an article followed by a noun: 50 Here we use the convention that the ﬁrst element of each list designates the part of speech for the rest of the words in the list. 572 (define (parse-noun-phrase) (list 'noun-phrase (parse-word articles) (parse-word nouns))) At the lowest level, parsing boils down to repeatedly checking that the next unparsed word is a member of the list of words for the required part of speech. To implement this, we maintain a global variable *unparsed*, which is the input that has not yet been parsed. Each time we check a word, we require that *unparsed* must be non-empty and that it should begin with a word from the designated list. If so, we remove that word from *unparsed* and return the word together with its part of speech (which is found at the head of the list):51 (define (parse-word word-list) (require (not (null? *unparsed*))) (require (memq (car *unparsed*) (cdr word-list))) (let ((found-word (car *unparsed*))) (set! *unparsed* (cdr *unparsed*)) (list (car word-list) found-word))) To start the parsing, all we need to do is set *unparsed* to be the entire input, try to parse a sentence, and check that nothing is le over: (define *unparsed* '()) (define (parse input) (set! *unparsed* input) (let ((sent (parse-sentence))) (require (null? *unparsed*)) sent)) We can now try the parser and verify that it works for our simple test sentence: 51 Noticethat parse-word uses set! to modify the unparsed input list. For this to work, our amb evaluator must undo the eﬀects of set! operations when it backtracks. 573 ;;; Amb-Eval input: (parse '(the cat eats)) ;;; Starting a new problem ;;; Amb-Eval value: (sentence (noun-phrase (article the) (noun cat)) (verb eats)) e amb evaluator is useful here because it is convenient to express the parsing constraints with the aid of require. Automatic search and backtracking really pay oﬀ, however, when we consider more complex grammars where there are choices for how the units can be decom- posed. Let’s add to our grammar a list of prepositions: (define prepositions '(prep for to in by with)) and deﬁne a prepositional phrase (e.g., “for the cat”) to be a preposition followed by a noun phrase: (define (parse-prepositional-phrase) (list 'prep-phrase (parse-word prepositions) (parse-noun-phrase))) Now we can deﬁne a sentence to be a noun phrase followed by a verb phrase, where a verb phrase can be either a verb or a verb phrase ex- tended by a prepositional phrase:52 (define (parse-sentence) (list 'sentence (parse-noun-phrase) (parse-verb-phrase))) (define (parse-verb-phrase) (define (maybe-extend verb-phrase) (amb verb-phrase 52 Observe that this deﬁnition is recursive—a verb may be followed by any number of prepositional phrases. 574 (maybe-extend (list 'verb-phrase verb-phrase (parse-prepositional-phrase))))) (maybe-extend (parse-word verbs))) While we’re at it, we can also elaborate the deﬁnition of noun phrases to permit such things as “a cat in the class.” What we used to call a noun phrase, we’ll now call a simple noun phrase, and a noun phrase will now be either a simple noun phrase or a noun phrase extended by a prepositional phrase: (define (parse-simple-noun-phrase) (list 'simple-noun-phrase (parse-word articles) (parse-word nouns))) (define (parse-noun-phrase) (define (maybe-extend noun-phrase) (amb noun-phrase (maybe-extend (list 'noun-phrase noun-phrase (parse-prepositional-phrase))))) (maybe-extend (parse-simple-noun-phrase))) Our new grammar lets us parse more complex sentences. For example (parse '(the student with the cat sleeps in the class)) produces (sentence (noun-phrase (simple-noun-phrase (article the) (noun student)) (prep-phrase (prep with) 575 (simple-noun-phrase (article the) (noun cat)))) (verb-phrase (verb sleeps) (prep-phrase (prep in) (simple-noun-phrase (article the) (noun class))))) Observe that a given input may have more than one legal parse. In the sentence “e professor lectures to the student with the cat,” it may be that the professor is lecturing with the cat, or that the student has the cat. Our nondeterministic program ﬁnds both possibilities: (parse '(the professor lectures to the student with the cat)) produces (sentence (simple-noun-phrase (article the) (noun professor)) (verb-phrase (verb-phrase (verb lectures) (prep-phrase (prep to) (simple-noun-phrase (article the) (noun student)))) (prep-phrase (prep with) (simple-noun-phrase (article the) (noun cat))))) Asking the evaluator to try again yields (sentence (simple-noun-phrase (article the) (noun professor)) (verb-phrase (verb lectures) (prep-phrase (prep to) 576 (noun-phrase (simple-noun-phrase (article the) (noun student)) (prep-phrase (prep with) (simple-noun-phrase (article the) (noun cat))))))) Exercise 4.45: With the grammar given above, the follow- ing sentence can be parsed in ﬁve diﬀerent ways: “e pro- fessor lectures to the student in the class with the cat.” Give the ﬁve parses and explain the diﬀerences in shades of mean- ing among them. Exercise 4.46: e evaluators in Section 4.1 and Section 4.2 do not determine what order operands are evaluated in. We will see that the amb evaluator evaluates them from le to right. Explain why our parsing program wouldn’t work if the operands were evaluated in some other order. Exercise 4.47: Louis Reasoner suggests that, since a verb phrase is either a verb or a verb phrase followed by a prepo- sitional phrase, it would be much more straightforward to deﬁne the procedure parse-verb-phrase as follows (and similarly for noun phrases): (define (parse-verb-phrase) (amb (parse-word verbs) (list 'verb-phrase (parse-verb-phrase) (parse-prepositional-phrase)))) Does this work? Does the program’s behavior change if we interchange the order of expressions in the amb? 577 Exercise 4.48: Extend the grammar given above to handle more complex sentences. For example, you could extend noun phrases and verb phrases to include adjectives and adverbs, or you could handle compound sentences.53 Exercise 4.49: Alyssa P. Hacker is more interested in gen- erating interesting sentences than in parsing them. She rea- sons that by simply changing the procedure parse-word so that it ignores the “input sentence” and instead always suc- ceeds and generates an appropriate word, we can use the programs we had built for parsing to do generation instead. Implement Alyssa’s idea, and show the ﬁrst half-dozen or so sentences generated.54 4.3.3 Implementing the amb Evaluator e evaluation of an ordinary Scheme expression may return a value, may never terminate, or may signal an error. In nondeterministic Scheme the evaluation of an expression may in addition result in the discovery of a dead end, in which case evaluation must backtrack to a previous 53 is kind of grammar can become arbitrarily complex, but it is only a toy as far as real language understanding is concerned. Real natural-language understand- ing by computer requires an elaborate mixture of syntactic analysis and interpretation of meaning. On the other hand, even toy parsers can be useful in supporting ﬂexi- ble command languages for programs such as information-retrieval systems. Winston 1992 discusses computational approaches to real language understanding and also the applications of simple grammars to command languages. 54 Although Alyssa’s idea works just ﬁne (and is surprisingly simple), the sentences that it generates are a bit boring—they don’t sample the possible sentences of this lan- guage in a very interesting way. In fact, the grammar is highly recursive in many places, and Alyssa’s technique “falls into” one of these recursions and gets stuck. See Exercise 4.50 for a way to deal with this. 578 choice point. e interpretation of nondeterministic Scheme is compli- cated by this extra case. We will construct the amb evaluator for nondeterministic Scheme by modifying the analyzing evaluator of Section 4.1.7.55 As in the analyz- ing evaluator, evaluation of an expression is accomplished by calling an execution procedure produced by analysis of that expression. e dif- ference between the interpretation of ordinary Scheme and the inter- pretation of nondeterministic Scheme will be entirely in the execution procedures. Execution procedures and continuations Recall that the execution procedures for the ordinary evaluator take one argument: the environment of execution. In contrast, the execution pro- cedures in the amb evaluator take three arguments: the environment, and two procedures called continuation procedures. e evaluation of an expression will ﬁnish by calling one of these two continuations: If the evaluation results in a value, the success continuation is called with that value; if the evaluation results in the discovery of a dead end, the failure continuation is called. Constructing and calling appropriate con- tinuations is the mechanism by which the nondeterministic evaluator implements backtracking. It is the job of the success continuation to receive a value and pro- ceed with the computation. Along with that value, the success contin- uation is passed another failure continuation, which is to be called sub- sequently if the use of that value leads to a dead end. 55 We chose to implement the lazy evaluator in Section 4.2 as a modiﬁcation of the ordinary metacircular evaluator of Section 4.1.1. In contrast, we will base the amb eval- uator on the analyzing evaluator of Section 4.1.7, because the execution procedures in that evaluator provide a convenient framework for implementing backtracking. 579 It is the job of the failure continuation to try another branch of the nondeterministic process. e essence of the nondeterministic language is in the fact that expressions may represent choices among alternatives. e evaluation of such an expression must proceed with one of the indi- cated alternative choices, even though it is not known in advance which choices will lead to acceptable results. To deal with this, the evaluator picks one of the alternatives and passes this value to the success con- tinuation. Together with this value, the evaluator constructs and passes along a failure continuation that can be called later to choose a diﬀerent alternative. A failure is triggered during evaluation (that is, a failure continua- tion is called) when a user program explicitly rejects the current line of aack (for example, a call to require may result in execution of (amb), an expression that always fails—see Section 4.3.1). e failure continu- ation in hand at that point will cause the most recent choice point to choose another alternative. If there are no more alternatives to be con- sidered at that choice point, a failure at an earlier choice point is trig- gered, and so on. Failure continuations are also invoked by the driver loop in response to a try-again request, to ﬁnd another value of the expression. In addition, if a side-eﬀect operation (such as assignment to a vari- able) occurs on a branch of the process resulting from a choice, it may be necessary, when the process ﬁnds a dead end, to undo the side eﬀect be- fore making a new choice. is is accomplished by having the side-eﬀect operation produce a failure continuation that undoes the side eﬀect and propagates the failure. In summary, failure continuations are constructed by • amb expressions—to provide a mechanism to make alternative choices if the current choice made by the amb expression leads to a dead end; 580 • the top-level driver—to provide a mechanism to report failure when the choices are exhausted; • assignments—to intercept failures and undo assignments during backtracking. Failures are initiated only when a dead end is encountered. is occurs • if the user program executes (amb); • if the user types try-again at the top-level driver. Failure continuations are also called during processing of a failure: • When the failure continuation created by an assignment ﬁnishes undoing a side eﬀect, it calls the failure continuation it inter- cepted, in order to propagate the failure back to the choice point that led to this assignment or to the top level. • When the failure continuation for an amb runs out of choices, it calls the failure continuation that was originally given to the amb, in order to propagate the failure back to the previous choice point or to the top level. Structure of the evaluator e syntax- and data-representation procedures for the amb evaluator, and also the basic analyze procedure, are identical to those in the eval- uator of Section 4.1.7, except for the fact that we need additional syntax procedures to recognize the amb special form:56 56 We assume that the evaluator supports let (see Exercise 4.22), which we have used in our nondeterministic programs. 581 (define (amb? exp) (tagged-list? exp 'amb)) (define (amb-choices exp) (cdr exp)) We must also add to the dispatch in analyze a clause that will recognize this special form and generate an appropriate execution procedure: ((amb? exp) (analyze-amb exp)) e top-level procedure ambeval (similar to the version of eval given in Section 4.1.7) analyzes the given expression and applies the resulting execution procedure to the given environment, together with two given continuations: (define (ambeval exp env succeed fail) ((analyze exp) env succeed fail)) A success continuation is a procedure of two arguments: the value just obtained and another failure continuation to be used if that value leads to a subsequent failure. A failure continuation is a procedure of no ar- guments. So the general form of an execution procedure is (lambda (env succeed fail) ;; succeed is (lambda (value fail) . . .) ;; fail is (lambda () . . .) . . .) For example, executing (ambeval ⟨exp⟩ the-global-environment (lambda (value fail) value) (lambda () 'failed)) will aempt to evaluate the given expression and will return either the expression’s value (if the evaluation succeeds) or the symbol failed (if the evaluation fails). e call to ambeval in the driver loop shown below 582 uses much more complicated continuation procedures, which continue the loop and support the try-again request. Most of the complexity of the amb evaluator results from the me- chanics of passing the continuations around as the execution proce- dures call each other. In going through the following code, you should compare each of the execution procedures with the corresponding pro- cedure for the ordinary evaluator given in Section 4.1.7. Simple expressions e execution procedures for the simplest kinds of expressions are es- sentially the same as those for the ordinary evaluator, except for the need to manage the continuations. e execution procedures simply succeed with the value of the expression, passing along the failure con- tinuation that was passed to them. (define (analyze-self-evaluating exp) (lambda (env succeed fail) (succeed exp fail))) (define (analyze-quoted exp) (let ((qval (text-of-quotation exp))) (lambda (env succeed fail) (succeed qval fail)))) (define (analyze-variable exp) (lambda (env succeed fail) (succeed (lookup-variable-value exp env) fail))) (define (analyze-lambda exp) (let ((vars (lambda-parameters exp)) (bproc (analyze-sequence (lambda-body exp)))) (lambda (env succeed fail) (succeed (make-procedure vars bproc env) fail)))) 583 Notice that looking up a variable always ‘succeeds.’ If lookup-variable- value fails to ﬁnd the variable, it signals an error, as usual. Such a “fail- ure” indicates a program bug—a reference to an unbound variable; it is not an indication that we should try another nondeterministic choice instead of the one that is currently being tried. Conditionals and sequences Conditionals are also handled in a similar way as in the ordinary eval- uator. e execution procedure generated by analyze-if invokes the predicate execution procedure pproc with a success continuation that checks whether the predicate value is true and goes on to execute ei- ther the consequent or the alternative. If the execution of pproc fails, the original failure continuation for the if expression is called. (define (analyze-if exp) (let ((pproc (analyze (if-predicate exp))) (cproc (analyze (if-consequent exp))) (aproc (analyze (if-alternative exp)))) (lambda (env succeed fail) (pproc env ;; success continuation for evaluating the predicate ;; to obtain pred-value (lambda (pred-value fail2) (if (true? pred-value) (cproc env succeed fail2) (aproc env succeed fail2))) ;; failure continuation for evaluating the predicate fail)))) Sequences are also handled in the same way as in the previous evaluator, except for the machinations in the subprocedure sequentially that are required for passing the continuations. Namely, to sequentially execute 584 a and then b, we call a with a success continuation that calls b. (define (analyze-sequence exps) (define (sequentially a b) (lambda (env succeed fail) (a env ;; success continuation for calling a (lambda (a-value fail2) (b env succeed fail2)) ;; failure continuation for calling a fail))) (define (loop first-proc rest-procs) (if (null? rest-procs) first-proc (loop (sequentially first-proc (car rest-procs)) (cdr rest-procs)))) (let ((procs (map analyze exps))) (if (null? procs) (error "Empty sequence: ANALYZE")) (loop (car procs) (cdr procs)))) Definitions and assignments Deﬁnitions are another case where we must go to some trouble to man- age the continuations, because it is necessary to evaluate the deﬁnition- value expression before actually deﬁning the new variable. To accom- plish this, the deﬁnition-value execution procedure vproc is called with the environment, a success continuation, and the failure continuation. If the execution of vproc succeeds, obtaining a value val for the deﬁned variable, the variable is deﬁned and the success is propagated: (define (analyze-definition exp) 585 (let ((var (definition-variable exp)) (vproc (analyze (definition-value exp)))) (lambda (env succeed fail) (vproc env (lambda (val fail2) (define-variable! var val env) (succeed 'ok fail2)) fail)))) Assignments are more interesting. is is the ﬁrst place where we really use the continuations, rather than just passing them around. e exe- cution procedure for assignments starts out like the one for deﬁnitions. It ﬁrst aempts to obtain the new value to be assigned to the variable. If this evaluation of vproc fails, the assignment fails. If vproc succeeds, however, and we go on to make the assignment, we must consider the possibility that this branch of the computation might later fail, which will require us to backtrack out of the assign- ment. us, we must arrange to undo the assignment as part of the backtracking process.57 is is accomplished by giving vproc a success continuation (marked with the comment “*1*” below) that saves the old value of the variable before assigning the new value to the variable and proceeding from the assignment. e failure continuation that is passed along with the value of the assignment (marked with the comment “*2*” below) restores the old value of the variable before continuing the failure. at is, a suc- cessful assignment provides a failure continuation that will intercept a subsequent failure; whatever failure would otherwise have called fail2 calls this procedure instead, to undo the assignment before actually call- ing fail2. 57 We didn’t worry about undoing deﬁnitions, since we can assume that internal def- initions are scanned out (Section 4.1.6). 586 (define (analyze-assignment exp) (let ((var (assignment-variable exp)) (vproc (analyze (assignment-value exp)))) (lambda (env succeed fail) (vproc env (lambda (val fail2) ; *1* (let ((old-value (lookup-variable-value var env))) (set-variable-value! var val env) (succeed 'ok (lambda () ; *2* (set-variable-value! var old-value env) (fail2))))) fail)))) Procedure applications e execution procedure for applications contains no new ideas except for the technical complexity of managing the continuations. is com- plexity arises in analyze-application, due to the need to keep track of the success and failure continuations as we evaluate the operands. We use a procedure get-args to evaluate the list of operands, rather than a simple map as in the ordinary evaluator. (define (analyze-application exp) (let ((fproc (analyze (operator exp))) (aprocs (map analyze (operands exp)))) (lambda (env succeed fail) (fproc env (lambda (proc fail2) (get-args aprocs env 587 (lambda (args fail3) (execute-application proc args succeed fail3)) fail2)) fail)))) In get-args, notice how cdr-ing down the list of aproc execution pro- cedures and consing up the resulting list of args is accomplished by calling each aproc in the list with a success continuation that recur- sively calls get-args. Each of these recursive calls to get-args has a success continuation whose value is the cons of the newly obtained ar- gument onto the list of accumulated arguments: (define (get-args aprocs env succeed fail) (if (null? aprocs) (succeed '() fail) ((car aprocs) env ;; success continuation for this aproc (lambda (arg fail2) (get-args (cdr aprocs) env ;; success continuation for ;; recursive call to get-args (lambda (args fail3) (succeed (cons arg args) fail3)) fail2)) fail))) e actual procedure application, which is performed by execute-appli- cation, is accomplished in the same way as for the ordinary evaluator, except for the need to manage the continuations. 588 (define (execute-application proc args succeed fail) (cond ((primitive-procedure? proc) (succeed (apply-primitive-procedure proc args) fail)) ((compound-procedure? proc) ((procedure-body proc) (extend-environment (procedure-parameters proc) args (procedure-environment proc)) succeed fail)) (else (error "Unknown procedure type: EXECUTE-APPLICATION" proc)))) Evaluating amb expressions e amb special form is the key element in the nondeterministic lan- guage. Here we see the essence of the interpretation process and the reason for keeping track of the continuations. e execution procedure for amb deﬁnes a loop try-next that cycles through the execution pro- cedures for all the possible values of the amb expression. Each execution procedure is called with a failure continuation that will try the next one. When there are no more alternatives to try, the entire amb expression fails. (define (analyze-amb exp) (let ((cprocs (map analyze (amb-choices exp)))) (lambda (env succeed fail) (define (try-next choices) (if (null? choices) (fail) 589 ((car choices) env succeed (lambda () (try-next (cdr choices)))))) (try-next cprocs)))) Driver loop e driver loop for the amb evaluator is complex, due to the mecha- nism that permits the user to try again in evaluating an expression. e driver uses a procedure called internal-loop, which takes as argument a procedure try-again. e intent is that calling try-again should go on to the next untried alternative in the nondeterministic evaluation. internal-loop either calls try-again in response to the user typing try-again at the driver loop, or else starts a new evaluation by calling ambeval. e failure continuation for this call to ambeval informs the user that there are no more values and re-invokes the driver loop. e success continuation for the call to ambeval is more subtle. We print the obtained value and then invoke the internal loop again with a try-again procedure that will be able to try the next alterna- tive. is next-alternative procedure is the second argument that was passed to the success continuation. Ordinarily, we think of this second argument as a failure continuation to be used if the current evaluation branch later fails. In this case, however, we have completed a successful evaluation, so we can invoke the “failure” alternative branch in order to search for additional successful evaluations. (define input-prompt ";;; Amb-Eval input:") (define output-prompt ";;; Amb-Eval value:") 590 (define (driver-loop) (define (internal-loop try-again) (prompt-for-input input-prompt) (let ((input (read))) (if (eq? input 'try-again) (try-again) (begin (newline) (display ";;; Starting a new problem ") (ambeval input the-global-environment ;; ambeval success (lambda (val next-alternative) (announce-output output-prompt) (user-print val) (internal-loop next-alternative)) ;; ambeval failure (lambda () (announce-output ";;; There are no more values of") (user-print input) (driver-loop))))))) (internal-loop (lambda () (newline) (display ";;; There is no current problem") (driver-loop)))) e initial call to internal-loop uses a try-again procedure that com- plains that there is no current problem and restarts the driver loop. is is the behavior that will happen if the user types try-again when there is no evaluation in progress. Exercise 4.50: Implement a new special form ramb that is 591 like amb except that it searches alternatives in a random or- der, rather than from le to right. Show how this can help with Alyssa’s problem in Exercise 4.49. Exercise 4.51: Implement a new kind of assignment called permanent-set! that is not undone upon failure. For ex- ample, we can choose two distinct elements from a list and count the number of trials required to make a successful choice as follows: (define count 0) (let ((x (an-element-of '(a b c))) (y (an-element-of '(a b c)))) (permanent-set! count (+ count 1)) (require (not (eq? x y))) (list x y count)) ;;; Starting a new problem ;;; Amb-Eval value: (a b 2) ;;; Amb-Eval input: try-again ;;; Amb-Eval value: (a c 3) What values would have been displayed if we had used set! here rather than permanent-set! ? Exercise 4.52: Implement a new construct called if-fail that permits the user to catch the failure of an expression. if-fail takes two expressions. It evaluates the ﬁrst expres- sion as usual and returns as usual if the evaluation suc- ceeds. If the evaluation fails, however, the value of the sec- ond expression is returned, as in the following example: 592 ;;; Amb-Eval input: (if-fail (let ((x (an-element-of '(1 3 5)))) (require (even? x)) x) 'all-odd) ;;; Starting a new problem ;;; Amb-Eval value: all-odd ;;; Amb-Eval input: (if-fail (let ((x (an-element-of '(1 3 5 8)))) (require (even? x)) x) 'all-odd) ;;; Starting a new problem ;;; Amb-Eval value: 8 Exercise 4.53: With permanent-set! as described in Exer- cise 4.51 and if-fail as in Exercise 4.52, what will be the result of evaluating (let ((pairs '())) (if-fail (let ((p (prime-sum-pair '(1 3 5 8) '(20 35 110)))) (permanent-set! pairs (cons p pairs)) (amb)) pairs)) Exercise 4.54: If we had not realized that require could be implemented as an ordinary procedure that uses amb, to be deﬁned by the user as part of a nondeterministic program, 593 we would have had to implement it as a special form. is would require syntax procedures (define (require? exp) (tagged-list? exp 'require)) (define (require-predicate exp) (cadr exp)) and a new clause in the dispatch in analyze ((require? exp) (analyze-require exp)) as well the procedure analyze-require that handles require expressions. Complete the following deﬁnition of analyze- require. (define (analyze-require exp) (let ((pproc (analyze (require-predicate exp)))) (lambda (env succeed fail) (pproc env (lambda (pred-value fail2) (if ⟨??⟩ ⟨??⟩ (succeed 'ok fail2))) fail)))) 4.4 Logic Programming In Chapter 1 we stressed that computer science deals with imperative (how to) knowledge, whereas mathematics deals with declarative (what is) knowledge. Indeed, programming languages require that the pro- grammer express knowledge in a form that indicates the step-by-step methods for solving particular problems. On the other hand, high-level 594 languages provide, as part of the language implementation, a substantial amount of methodological knowledge that frees the user from concern with numerous details of how a speciﬁed computation will progress. Most programming languages, including Lisp, are organized around computing the values of mathematical functions. Expression-oriented languages (such as Lisp, Fortran, and Algol) capitalize on the “pun” that an expression that describes the value of a function may also be inter- preted as a means of computing that value. Because of this, most pro- gramming languages are strongly biased toward unidirectional compu- tations (computations with well-deﬁned inputs and outputs). ere are, however, radically diﬀerent programming languages that relax this bias. We saw one such example in Section 3.3.5, where the objects of compu- tation were arithmetic constraints. In a constraint system the direction and the order of computation are not so well speciﬁed; in carrying out a computation the system must therefore provide more detailed “how to” knowledge than would be the case with an ordinary arithmetic compu- tation. is does not mean, however, that the user is released altogether from the responsibility of providing imperative knowledge. ere are many constraint networks that implement the same set of constraints, and the user must choose from the set of mathematically equivalent networks a suitable network to specify a particular computation. e nondeterministic program evaluator of Section 4.3 also moves away from the view that programming is about constructing algorithms for computing unidirectional functions. In a nondeterministic language, expressions can have more than one value, and, as a result, the compu- tation is dealing with relations rather than with single-valued functions. Logic programming extends this idea by combining a relational vision of programming with a powerful kind of symbolic paern matching 595 called uniﬁcation.58 is approach, when it works, can be a very powerful way to write programs. Part of the power comes from the fact that a single “what is” fact can be used to solve a number of diﬀerent problems that would have diﬀerent “how to” components. As an example, consider the append op- eration, which takes two lists as arguments and combines their elements to form a single list. In a procedural language such as Lisp, we could deﬁne append in terms of the basic list constructor cons, as we did in Section 2.2.1: 58 Logic programming has grown out of a long history of research in automatic the- orem proving. Early theorem-proving programs could accomplish very lile, because they exhaustively searched the space of possible proofs. e major breakthrough that made such a search plausible was the discovery in the early 1960s of the uniﬁcation algorithm and the resolution principle (Robinson 1965). Resolution was used, for exam- ple, by Green and Raphael (1968) (see also Green 1969) as the basis for a deductive question-answering system. During most of this period, researchers concentrated on algorithms that are guaranteed to ﬁnd a proof if one exists. Such algorithms were dif- ﬁcult to control and to direct toward a proof. Hewi (1969) recognized the possibility of merging the control structure of a programming language with the operations of a logic-manipulation system, leading to the work in automatic search mentioned in Sec- tion 4.3.1 (Footnote 4.47). At the same time that this was being done, Colmerauer, in Marseille, was developing rule-based systems for manipulating natural language (see Colmerauer et al. 1973). He invented a programming language called Prolog for repre- senting those rules. Kowalski (1973; 1979), in Edinburgh, recognized that execution of a Prolog program could be interpreted as proving theorems (using a proof technique called linear Horn-clause resolution). e merging of the last two strands led to the logic-programming movement. us, in assigning credit for the development of logic programming, the French can point to Prolog’s genesis at the University of Marseille, while the British can highlight the work at the University of Edinburgh. According to people at , logic programming was developed by these groups in an aempt to ﬁg- ure out what Hewi was talking about in his brilliant but impenetrable Ph.D. thesis. For a history of logic programming, see Robinson 1983. 596 (define (append x y) (if (null? x) y (cons (car x) (append (cdr x) y)))) is procedure can be regarded as a translation into Lisp of the follow- ing two rules, the ﬁrst of which covers the case where the ﬁrst list is empty and the second of which handles the case of a nonempty list, which is a cons of two parts: • For any list y, the empty list and y append to form y. • For any u, v, y, and z, (cons u v) and y append to form (cons u z) if v and y append to form z.59 Using the append procedure, we can answer questions such as Find the append of (a b) and (c d). But the same two rules are also suﬃcient for answering the following sorts of questions, which the procedure can’t answer: Find a list y that appends with (a b) to produce (a b c d). Find all x and y that append to form (a b c d). In a logic programming language, the programmer writes an append “procedure” by stating the two rules about append given above. “How to” knowledge is provided automatically by the interpreter to allow this 59 To see the correspondence between the rules and the procedure, let x in the pro- cedure (where x is nonempty) correspond to (cons u v) in the rule. en z in the rule corresponds to the append of (cdr x) and y. 597 single pair of rules to be used to answer all three types of questions about append.60 Contemporary logic programming languages (including the one we implement here) have substantial deﬁciencies, in that their general “how to” methods can lead them into spurious inﬁnite loops or other unde- sirable behavior. Logic programming is an active ﬁeld of research in computer science.61 Earlier in this chapter we explored the technology of implementing interpreters and described the elements that are essential to an inter- preter for a Lisp-like language (indeed, to an interpreter for any con- ventional language). Now we will apply these ideas to discuss an in- terpreter for a logic programming language. We call this language the query language, because it is very useful for retrieving information from data bases by formulating queries, or questions, expressed in the lan- guage. Even though the query language is very diﬀerent from Lisp, we 60 is certainly does not relieve the user of the entire problem of how to compute the answer. ere are many diﬀerent mathematically equivalent sets of rules for for- mulating the append relation, only some of which can be turned into eﬀective devices for computing in any direction. In addition, sometimes “what is” information gives no clue “how to” compute an answer. For example, consider the problem of computing the y such that y 2 = x . 61 Interest in logic programming peaked during the early 80s when the Japanese gov- ernment began an ambitious project aimed at building superfast computers optimized to run logic programming languages. e speed of such computers was to be measured in LIPS (Logical Inferences Per Second) rather than the usual FLOPS (FLoating-point Operations Per Second). Although the project succeeded in developing hardware and soware as originally planned, the international computer industry moved in a dif- ferent direction. See Feigenbaum and Shrobe 1993 for an overview evaluation of the Japanese project. e logic programming community has also moved on to consider relational programming based on techniques other than simple paern matching, such as the ability to deal with numerical constraints such as the ones illustrated in the constraint-propagation system of Section 3.3.5. 598 will ﬁnd it convenient to describe the language in terms of the same gen- eral framework we have been using all along: as a collection of primitive elements, together with means of combination that enable us to com- bine simple elements to create more complex elements and means of ab- straction that enable us to regard complex elements as single conceptual units. An interpreter for a logic programming language is considerably more complex than an interpreter for a language like Lisp. Neverthe- less, we will see that our query-language interpreter contains many of the same elements found in the interpreter of Section 4.1. In particu- lar, there will be an “eval” part that classiﬁes expressions according to type and an “apply” part that implements the language’s abstraction mechanism (procedures in the case of Lisp, and rules in the case of logic programming). Also, a central role is played in the implementation by a frame data structure, which determines the correspondence between symbols and their associated values. One additional interesting aspect of our query-language implementation is that we make substantial use of streams, which were introduced in Chapter 3. 4.4.1 Deductive Information Retrieval Logic programming excels in providing interfaces to data bases for in- formation retrieval. e query language we shall implement in this chap- ter is designed to be used in this way. In order to illustrate what the query system does, we will show how it can be used to manage the data base of personnel records for Mi- crosha, a thriving high-technology company in the Boston area. e language provides paern-directed access to personnel information and can also take advantage of general rules in order to make logical deduc- tions. 599 A sample data base e personnel data base for Microsha contains assertions about com- pany personnel. Here is the information about Ben Bitdiddle, the resi- dent computer wizard: (address (Bitdiddle Ben) (Slumerville (Ridge Road) 10)) (job (Bitdiddle Ben) (computer wizard)) (salary (Bitdiddle Ben) 60000) Each assertion is a list (in this case a triple) whose elements can them- selves be lists. As resident wizard, Ben is in charge of the company’s computer division, and he supervises two programmers and one technician. Here is the information about them: (address (Hacker Alyssa P) (Cambridge (Mass Ave) 78)) (job (Hacker Alyssa P) (computer programmer)) (salary (Hacker Alyssa P) 40000) (supervisor (Hacker Alyssa P) (Bitdiddle Ben)) (address (Fect Cy D) (Cambridge (Ames Street) 3)) (job (Fect Cy D) (computer programmer)) (salary (Fect Cy D) 35000) (supervisor (Fect Cy D) (Bitdiddle Ben)) (address (Tweakit Lem E) (Boston (Bay State Road) 22)) (job (Tweakit Lem E) (computer technician)) (salary (Tweakit Lem E) 25000) (supervisor (Tweakit Lem E) (Bitdiddle Ben)) ere is also a programmer trainee, who is supervised by Alyssa: (address (Reasoner Louis) (Slumerville (Pine Tree Road) 80)) (job (Reasoner Louis) (computer programmer trainee)) 600 (salary (Reasoner Louis) 30000) (supervisor (Reasoner Louis) (Hacker Alyssa P)) All of these people are in the computer division, as indicated by the word computer as the ﬁrst item in their job descriptions. Ben is a high-level employee. His supervisor is the company’s big wheel himself: (supervisor (Bitdiddle Ben) (Warbucks Oliver)) (address (Warbucks Oliver) (Swellesley (Top Heap Road))) (job (Warbucks Oliver) (administration big wheel)) (salary (Warbucks Oliver) 150000) Besides the computer division supervised by Ben, the company has an accounting division, consisting of a chief accountant and his assistant: (address (Scrooge Eben) (Weston (Shady Lane) 10)) (job (Scrooge Eben) (accounting chief accountant)) (salary (Scrooge Eben) 75000) (supervisor (Scrooge Eben) (Warbucks Oliver)) (address (Cratchet Robert) (Allston (N Harvard Street) 16)) (job (Cratchet Robert) (accounting scrivener)) (salary (Cratchet Robert) 18000) (supervisor (Cratchet Robert) (Scrooge Eben)) ere is also a secretary for the big wheel: (address (Aull DeWitt) (Slumerville (Onion Square) 5)) (job (Aull DeWitt) (administration secretary)) (salary (Aull DeWitt) 25000) (supervisor (Aull DeWitt) (Warbucks Oliver)) e data base also contains assertions about which kinds of jobs can be done by people holding other kinds of jobs. For instance, a computer 601 wizard can do the jobs of both a computer programmer and a computer technician: (can-do-job (computer wizard) (computer programmer)) (can-do-job (computer wizard) (computer technician)) A computer programmer could ﬁll in for a trainee: (can-do-job (computer programmer) (computer programmer trainee)) Also, as is well known, (can-do-job (administration secretary) (administration big wheel)) Simple queries e query language allows users to retrieve information from the data base by posing queries in response to the system’s prompt. For example, to ﬁnd all computer programmers one can say ;;; Query input: (job ?x (computer programmer)) e system will respond with the following items: ;;; Query results: (job (Hacker Alyssa P) (computer programmer)) (job (Fect Cy D) (computer programmer)) e input query speciﬁes that we are looking for entries in the data base that match a certain paern. In this example, the paern speciﬁes en- tries consisting of three items, of which the ﬁrst is the literal symbol job, the second can be anything, and the third is the literal list (computer programmer). e “anything” that can be the second item in the match- ing list is speciﬁed by a paern variable, ?x. e general form of a paern 602 variable is a symbol, taken to be the name of the variable, preceded by a question mark. We will see below why it is useful to specify names for paern variables rather than just puing ? into paerns to repre- sent “anything.” e system responds to a simple query by showing all entries in the data base that match the speciﬁed paern. A paern can have more than one variable. For example, the query (address ?x ?y) will list all the employees’ addresses. A paern can have no variables, in which case the query simply determines whether that paern is an entry in the data base. If so, there will be one match; if not, there will be no matches. e same paern variable can appear more than once in a query, specifying that the same “anything” must appear in each position. is is why variables have names. For example, (supervisor ?x ?x) ﬁnds all people who supervise themselves (though there are no such assertions in our sample data base). e query (job ?x (computer ?type)) matches all job entries whose third item is a two-element list whose ﬁrst item is computer: (job (Bitdiddle Ben) (computer wizard)) (job (Hacker Alyssa P) (computer programmer)) (job (Fect Cy D) (computer programmer)) (job (Tweakit Lem E) (computer technician)) is same paern does not match (job (Reasoner Louis) (computer programmer trainee)) 603 because the third item in the entry is a list of three elements, and the paern’s third item speciﬁes that there should be two elements. If we wanted to change the paern so that the third item could be any list beginning with computer, we could specify62 (job ?x (computer . ?type)) For example, (computer . ?type) matches the data (computer programmer trainee) with ?type as the list (programmer trainee). It also matches the data (computer programmer) with ?type as the list (programmer), and matches the data (computer) with ?type as the empty list (). We can describe the query language’s processing of simple queries as follows: • e system ﬁnds all assignments to variables in the query paern that satisfy the paern—that is, all sets of values for the variables such that if the paern variables are instantiated with (replaced by) the values, the result is in the data base. • e system responds to the query by listing all instantiations of the query paern with the variable assignments that satisfy it. 62 is uses the doed-tail notation introduced in Exercise 2.20. 604 Note that if the paern has no variables, the query reduces to a deter- mination of whether that paern is in the data base. If so, the empty assignment, which assigns no values to variables, satisﬁes that paern for that data base. Exercise 4.55: Give simple queries that retrieve the follow- ing information from the data base: 1. all people supervised by Ben Bitdiddle; 2. the names and jobs of all people in the accounting di- vision; 3. the names and addresses of all people who live in Slumerville. Compound queries Simple queries form the primitive operations of the query language. In order to form compound operations, the query language provides means of combination. One thing that makes the query language a logic programming language is that the means of combination mirror the means of combination used in forming logical expressions: and, or, and not. (Here and, or, and not are not the Lisp primitives, but rather oper- ations built into the query language.) We can use and as follows to ﬁnd the addresses of all the computer programmers: (and (job ?person (computer programmer)) (address ?person ?where)) e resulting output is (and (job (Hacker Alyssa P) (computer programmer)) (address (Hacker Alyssa P) (Cambridge (Mass Ave) 78))) 605 (and (job (Fect Cy D) (computer programmer)) (address (Fect Cy D) (Cambridge (Ames Street) 3))) In general, (and ⟨query1 ⟩ ⟨query2 ⟩ . . . ⟨queryn ⟩) is satisﬁed by all sets of values for the paern variables that simultane- ously satisfy ⟨query 1 ⟩ . . . ⟨queryn ⟩. As for simple queries, the system processes a compound query by ﬁnding all assignments to the paern variables that satisfy the query, then displaying instantiations of the query with those values. Another means of constructing compound queries is through or. For example, (or (supervisor ?x (Bitdiddle Ben)) (supervisor ?x (Hacker Alyssa P))) will ﬁnd all employees supervised by Ben Bitdiddle or Alyssa P. Hacker: (or (supervisor (Hacker Alyssa P) (Bitdiddle Ben)) (supervisor (Hacker Alyssa P) (Hacker Alyssa P))) (or (supervisor (Fect Cy D) (Bitdiddle Ben)) (supervisor (Fect Cy D) (Hacker Alyssa P))) (or (supervisor (Tweakit Lem E) (Bitdiddle Ben)) (supervisor (Tweakit Lem E) (Hacker Alyssa P))) (or (supervisor (Reasoner Louis) (Bitdiddle Ben)) (supervisor (Reasoner Louis) (Hacker Alyssa P))) In general, (or ⟨query1 ⟩ ⟨query2 ⟩ . . . ⟨queryn ⟩) is satisﬁed by all sets of values for the paern variables that satisfy at least one of ⟨query 1 ⟩ . . . ⟨queryn ⟩. Compound queries can also be formed with not. For example, 606 (and (supervisor ?x (Bitdiddle Ben)) (not (job ?x (computer programmer)))) ﬁnds all people supervised by Ben Bitdiddle who are not computer pro- grammers. In general, (not ⟨query1 ⟩) is satisﬁed by all assignments to the paern variables that do not satisfy ⟨query 1 ⟩.63 e ﬁnal combining form is called lisp-value. When lisp-value is the ﬁrst element of a paern, it speciﬁes that the next element is a Lisp predicate to be applied to the rest of the (instantiated) elements as arguments. In general, (lisp-value ⟨predicate⟩ ⟨arg1 ⟩ . . . ⟨argn ⟩) will be satisﬁed by assignments to the paern variables for which the ⟨predicate⟩ applied to the instantiated ⟨arg 1 ⟩ . . . ⟨argn ⟩ is true. For ex- ample, to ﬁnd all people whose salary is greater than $30,000 we could write64 (and (salary ?person ?amount) (lisp-value > ?amount 30000)) Exercise 4.56: Formulate compound queries that retrieve the following information: 63 Actually, this description of not is valid only for simple cases. e real behavior of not is more complex. We will examine not’s peculiarities in sections Section 4.4.2 and Section 4.4.3. 64 lisp-value should be used only to perform an operation not provided in the query language. In particular, it should not be used to test equality (since that is what the matching in the query language is designed to do) or inequality (since that can be done with the same rule shown below). 607 a. the names of all people who are supervised by Ben Bitdiddle, together with their addresses; b. all people whose salary is less than Ben Bitdiddle’s, together with their salary and Ben Bitdiddle’s salary; c. all people who are supervised by someone who is not in the computer division, together with the supervi- sor’s name and job. Rules In addition to primitive queries and compound queries, the query lan- guage provides means for abstracting queries. ese are given by rules. e rule (rule (lives-near ?person-1 ?person-2) (and (address ?person-1 (?town . ?rest-1)) (address ?person-2 (?town . ?rest-2)) (not (same ?person-1 ?person-2)))) speciﬁes that two people live near each other if they live in the same town. e ﬁnal not clause prevents the rule from saying that all peo- ple live near themselves. e same relation is deﬁned by a very simple rule:65 (rule (same ?x ?x)) 65 Notice that we do not need same in order to make two things be the same: We just use the same paern variable for each—in eﬀect, we have one thing instead of two things in the ﬁrst place. For example, see ?town in the lives-near rule and ?middle- manager in the wheel rule below. same is useful when we want to force two things to be diﬀerent, such as ?person-1 and ?person-2 in the lives-near rule. Although using the same paern variable in two parts of a query forces the same value to appear in both places, using diﬀerent paern variables does not force diﬀerent values to appear. (e values assigned to diﬀerent paern variables may be the same or diﬀerent.) 608 e following rule declares that a person is a “wheel” in an organization if he supervises someone who is in turn a supervisor: (rule (wheel ?person) (and (supervisor ?middle-manager ?person) (supervisor ?x ?middle-manager))) e general form of a rule is (rule ⟨conclusion⟩ ⟨body⟩) where ⟨conclusion⟩ is a paern and ⟨body ⟩ is any query.66 We can think of a rule as representing a large (even inﬁnite) set of assertions, namely all instantiations of the rule conclusion with variable assignments that satisfy the rule body. When we described simple queries (paerns), we said that an assignment to variables satisﬁes a paern if the instantiated paern is in the data base. But the paern needn’t be explicitly in the data base as an assertion. It can be an implicit assertion implied by a rule. For example, the query (lives-near ?x (Bitdiddle Ben)) results in (lives-near (Reasoner Louis) (Bitdiddle Ben)) (lives-near (Aull DeWitt) (Bitdiddle Ben)) To ﬁnd all computer programmers who live near Ben Bitdiddle, we can ask (and (job ?x (computer programmer)) (lives-near ?x (Bitdiddle Ben))) 66 We will also allow rules without bodies, as in same, and we will interpret such a rule to mean that the rule conclusion is satisﬁed by any values of the variables. 609 As in the case of compound procedures, rules can be used as parts of other rules (as we saw with the lives-near rule above) or even be de- ﬁned recursively. For instance, the rule (rule (outranked-by ?staff-person ?boss) (or (supervisor ?staff-person ?boss) (and (supervisor ?staff-person ?middle-manager) (outranked-by ?middle-manager ?boss)))) says that a staﬀ person is outranked by a boss in the organization if the boss is the person’s supervisor or (recursively) if the person’s supervisor is outranked by the boss. Exercise 4.57: Deﬁne a rule that says that person 1 can re- place person 2 if either person 1 does the same job as person 2 or someone who does person 1’s job can also do person 2’s job, and if person 1 and person 2 are not the same person. Using your rule, give queries that ﬁnd the following: a. all people who can replace Cy D. Fect; b. all people who can replace someone who is being paid more than they are, together with the two salaries. Exercise 4.58: Deﬁne a rule that says that a person is a “big shot” in a division if the person works in the division but does not have a supervisor who works in the division. Exercise 4.59: Ben Bitdiddle has missed one meeting too many. Fearing that his habit of forgeing meetings could cost him his job, Ben decides to do something about it. He adds all the weekly meetings of the ﬁrm to the Microsha data base by asserting the following: 610 (meeting accounting (Monday 9am)) (meeting administration (Monday 10am)) (meeting computer (Wednesday 3pm)) (meeting administration (Friday 1pm)) Each of the above assertions is for a meeting of an entire di- vision. Ben also adds an entry for the company-wide meet- ing that spans all the divisions. All of the company’s em- ployees aend this meeting. (meeting whole-company (Wednesday 4pm)) a. On Friday morning, Ben wants to query the data base for all the meetings that occur that day. What query should he use? b. Alyssa P. Hacker is unimpressed. She thinks it would be much more useful to be able to ask for her meetings by specifying her name. So she designs a rule that says that a person’s meetings include all whole-company meetings plus all meetings of that person’s division. Fill in the body of Alyssa’s rule. (rule (meeting-time ?person ?day-and-time) ⟨rule-body⟩) c. Alyssa arrives at work on Wednesday morning and wonders what meetings she has to aend that day. Having deﬁned the above rule, what query should she make to ﬁnd this out? Exercise 4.60: By giving the query (lives-near ?person (Hacker Alyssa P)) 611 Alyssa P. Hacker is able to ﬁnd people who live near her, with whom she can ride to work. On the other hand, when she tries to ﬁnd all pairs of people who live near each other by querying (lives-near ?person-1 ?person-2) she notices that each pair of people who live near each other is listed twice; for example, (lives-near (Hacker Alyssa P) (Fect Cy D)) (lives-near (Fect Cy D) (Hacker Alyssa P)) Why does this happen? Is there a way to ﬁnd a list of people who live near each other, in which each pair appears only once? Explain. Logic as programs We can regard a rule as a kind of logical implication: If an assignment of values to paern variables satisﬁes the body, then it satisﬁes the con- clusion. Consequently, we can regard the query language as having the ability to perform logical deductions based upon the rules. As an exam- ple, consider the append operation described at the beginning of Section 4.4. As we said, append can be characterized by the following two rules: • For any list y, the empty list and y append to form y. • For any u, v, y, and z, (cons u v) and y append to form (cons u z) if v and y append to form z. To express this in our query language, we deﬁne two rules for a relation (append-to-form x y z) 612 which we can interpret to mean “x and y append to form z”: (rule (append-to-form () ?y ?y)) (rule (append-to-form (?u . ?v) ?y (?u . ?z)) (append-to-form ?v ?y ?z)) e ﬁrst rule has no body, which means that the conclusion holds for any value of ?y. Note how the second rule makes use of doed-tail no- tation to name the car and cdr of a list. Given these two rules, we can formulate queries that compute the append of two lists: ;;; Query input: (append-to-form (a b) (c d) ?z) ;;; Query results: (append-to-form (a b) (c d) (a b c d)) What is more striking, we can use the same rules to ask the question “Which list, when appended to (a b), yields (a b c d)?” is is done as follows: ;;; Query input: (append-to-form (a b) ?y (a b c d)) ;;; Query results: (append-to-form (a b) (c d) (a b c d)) We can also ask for all pairs of lists that append to form (a b c d): ;;; Query input: (append-to-form ?x ?y (a b c d)) ;;; Query results: (append-to-form () (a b c d) (a b c d)) (append-to-form (a) (b c d) (a b c d)) (append-to-form (a b) (c d) (a b c d)) (append-to-form (a b c) (d) (a b c d)) (append-to-form (a b c d) () (a b c d)) 613 e query system may seem to exhibit quite a bit of intelligence in using the rules to deduce the answers to the queries above. Actually, as we will see in the next section, the system is following a well-determined algorithm in unraveling the rules. Unfortunately, although the system works impressively in the append case, the general methods may break down in more complex cases, as we will see in Section 4.4.3. Exercise 4.61: e following rules implement a next-to relation that ﬁnds adjacent elements of a list: (rule (?x next-to ?y in (?x ?y . ?u))) (rule (?x next-to ?y in (?v . ?z)) (?x next-to ?y in ?z)) What will the response be to the following queries? (?x next-to ?y in (1 (2 3) 4)) (?x next-to 1 in (2 1 3 1)) Exercise 4.62: Deﬁne rules to implement the last-pair operation of Exercise 2.17, which returns a list containing the last element of a nonempty list. Check your rules on queries such as (last-pair (3) ?x), (last-pair (1 2 3) ?x) and (last-pair (2 ?x) (3)). Do your rules work correctly on queries such as (last-pair ?x (3)) ? Exercise 4.63: e following data base (see Genesis 4) traces the genealogy of the descendants of Ada back to Adam, by way of Cain: (son Adam Cain) (son Cain Enoch) (son Enoch Irad) 614 (son Irad Mehujael) (son Mehujael Methushael) (son Methushael Lamech) (wife Lamech Ada) (son Ada Jabal) (son Ada Jubal) Formulate rules such as “If S is the son of f , and f is the son of G, then S is the grandson of G” and “If W is the wife of M , and S is the son of W , then S is the son of M ” (which was supposedly more true in biblical times than today) that will enable the query system to ﬁnd the grandson of Cain; the sons of Lamech; the grandsons of Methushael. (See Ex- ercise 4.69 for some rules to deduce more complicated re- lationships.) 4.4.2 How the ery System Works In Section 4.4.4 we will present an implementation of the query inter- preter as a collection of procedures. In this section we give an overview that explains the general structure of the system independent of low- level implementation details. Aer describing the implementation of the interpreter, we will be in a position to understand some of its limitations and some of the subtle ways in which the query language’s logical op- erations diﬀer from the operations of mathematical logic. It should be apparent that the query evaluator must perform some kind of search in order to match queries against facts and rules in the data base. One way to do this would be to implement the query system as a nondeterministic program, using the amb evaluator of Section 4.3 (see Exercise 4.78). Another possibility is to manage the search with the aid of streams. Our implementation follows this second approach. 615 e query system is organized around two central operations called paern matching and uniﬁcation. We ﬁrst describe paern matching and explain how this operation, together with the organization of informa- tion in terms of streams of frames, enables us to implement both simple and compound queries. We next discuss uniﬁcation, a generalization of paern matching needed to implement rules. Finally, we show how the entire query interpreter ﬁts together through a procedure that classiﬁes expressions in a manner analogous to the way eval classiﬁes expres- sions for the interpreter described in Section 4.1. Paern matching A paern matcher is a program that tests whether some datum ﬁts a speciﬁed paern. For example, the data list ((a b) c (a b)) matches the paern (?x c ?x) with the paern variable ?x bound to (a b). e same data list matches the paern (?x ?y ?z) with ?x and ?z both bound to (a b) and ?y bound to c. It also matches the paern ((?x ?y) c (?x ?y)) with ?x bound to a and ?y bound to b. However, it does not match the paern (?x a ?y), since that paern speciﬁes a list whose second element is the symbol a. e paern matcher used by the query system takes as inputs a paern, a datum, and a frame that speciﬁes bindings for various paern variables. It checks whether the datum matches the paern in a way that is consistent with the bindings already in the frame. If so, it returns the given frame augmented by any bindings that may have been determined by the match. Otherwise, it indicates that the match has failed. For example, using the paern (?x ?y ?x) to match (a b a) given an empty frame will return a frame specifying that ?x is bound to a and ?y is bound to b. Trying the match with the same paern, the same datum, and a frame specifying that ?y is bound to a will fail. Trying the 616 match with the same paern, the same datum, and a frame in which ?y is bound to b and ?x is unbound will return the given frame augmented by a binding of ?x to a. e paern matcher is all the mechanism that is needed to pro- cess simple queries that don’t involve rules. For instance, to process the query (job ?x (computer programmer)) we scan through all assertions in the data base and select those that match the paern with respect to an initially empty frame. For each match we ﬁnd, we use the frame returned by the match to instantiate the paern with a value for ?x. Streams of frames e testing of paerns against frames is organized through the use of streams. Given a single frame, the matching process runs through the data-base entries one by one. For each data-base entry, the matcher gen- erates either a special symbol indicating that the match has failed or an extension to the frame. e results for all the data-base entries are col- lected into a stream, which is passed through a ﬁlter to weed out the failures. e result is a stream of all the frames that extend the given frame via a match to some assertion in the data base.67 67 Because matching is generally very expensive, we would like to avoid applying the full matcher to every element of the data base. is is usually arranged by breaking up the process into a fast, coarse match and the ﬁnal match. e coarse match ﬁlters the data base to produce a small set of candidates for the ﬁnal match. With care, we can arrange our data base so that some of the work of coarse matching can be done when the data base is constructed rather then when we want to select the candidates. is is called indexing the data base. ere is a vast technology built around data-base- indexing schemes. Our implementation, described in Section 4.4.4, contains a simple- minded form of such an optimization. 617 input stream output stream of frames, of frames query filtered and extended (job ?x ?y) stream of assertions from data base Figure 4.4: A query processes a stream of frames. In our system, a query takes an input stream of frames and per- forms the above matching operation for every frame in the stream, as indicated in Figure 4.4. at is, for each frame in the input stream, the query generates a new stream consisting of all extensions to that frame by matches to assertions in the data base. All these streams are then combined to form one huge stream, which contains all possible exten- sions of every frame in the input stream. is stream is the output of the query. To answer a simple query, we use the query with an input stream consisting of a single empty frame. e resulting output stream contains all extensions to the empty frame (that is, all answers to our query). is stream of frames is then used to generate a stream of copies of the original query paern with the variables instantiated by the values in each frame, and this is the stream that is ﬁnally printed. Compound queries e real elegance of the stream-of-frames implementation is evident when we deal with compound queries. e processing of compound 618 input stream (and A B) output stream of frames of frames A B data base Figure 4.5: e and combination of two queries is produced by operating on the stream of frames in series. queries makes use of the ability of our matcher to demand that a match be consistent with a speciﬁed frame. For example, to handle the and of two queries, such as (and (can-do-job ?x (computer programmer trainee)) (job ?person ?x)) (informally, “Find all people who can do the job of a computer program- mer trainee”), we ﬁrst ﬁnd all entries that match the paern (can-do-job ?x (computer programmer trainee)) is produces a stream of frames, each of which contains a binding for ?x. en for each frame in the stream we ﬁnd all entries that match (job ?person ?x) in a way that is consistent with the given binding for ?x. Each such match will produce a frame containing bindings for ?x and ?person. e and of two queries can be viewed as a series combination of the two component queries, as shown in Figure 4.5. e frames that pass 619 (or A B) A input stream output stream of frames of frames merge B data base Figure 4.6: e or combination of two queries is produced by operating on the stream of frames in parallel and merg- ing the results. through the ﬁrst query ﬁlter are ﬁltered and further extended by the second query. Figure 4.6 shows the analogous method for computing the or of two queries as a parallel combination of the two component queries. e input stream of frames is extended separately by each query. e two resulting streams are then merged to produce the ﬁnal output stream. Even from this high-level description, it is apparent that the pro- cessing of compound queries can be slow. For example, since a query may produce more than one output frame for each input frame, and each query in an and gets its input frames from the previous query, an and query could, in the worst case, have to perform a number of matches 620 that is exponential in the number of queries (see Exercise 4.76).68 ough systems for handling only simple queries are quite practical, dealing with complex queries is extremely diﬃcult.69 From the stream-of-frames viewpoint, the not of some query acts as a ﬁlter that removes all frames for which the query can be satisﬁed. For instance, given the paern (not (job ?x (computer programmer))) we aempt, for each frame in the input stream, to produce extension frames that satisfy (job ?x (computer programmer)). We remove from the input stream all frames for which such extensions exist. e result is a stream consisting of only those frames in which the binding for ?x does not satisfy (job ?x (computer programmer)). For example, in processing the query (and (supervisor ?x ?y) (not (job ?x (computer programmer)))) the ﬁrst clause will generate frames with bindings for ?x and ?y. e not clause will then ﬁlter these by removing all frames in which the binding for ?x satisﬁes the restriction that ?x is a computer programmer.70 e lisp-value special form is implemented as a similar ﬁlter on frame streams. We use each frame in the stream to instantiate any vari- ables in the paern, then apply the Lisp predicate. We remove from the input stream all frames for which the predicate fails. 68 But this kind of exponential explosion is not common in and queries because the added conditions tend to reduce rather than expand the number of frames produced. 69 ere is a large literature on data-base-management systems that is concerned with how to handle complex queries eﬃciently. 70 ere is a subtle diﬀerence between this ﬁlter implementation of not and the usual meaning of not in mathematical logic. See Section 4.4.3. 621 Unification In order to handle rules in the query language, we must be able to ﬁnd the rules whose conclusions match a given query paern. Rule conclu- sions are like assertions except that they can contain variables, so we will need a generalization of paern matching—called uniﬁcation—in which both the “paern” and the “datum” may contain variables. A uniﬁer takes two paerns, each containing constants and vari- ables, and determines whether it is possible to assign values to the vari- ables that will make the two paerns equal. If so, it returns a frame containing these bindings. For example, unifying (?x a ?y) and (?y ?z a) will specify a frame in which ?x, ?y, and ?z must all be bound to a. On the other hand, unifying (?x ?y a) and (?x b ?y) will fail, because there is no value for ?y that can make the two paerns equal. (For the second elements of the paerns to be equal, ?y would have to be b; however, for the third elements to be equal, ?y would have to be a.) e uniﬁer used in the query system, like the paern matcher, takes a frame as input and performs uniﬁcations that are consistent with this frame. e uniﬁcation algorithm is the most technically diﬃcult part of the query system. With complex paerns, performing uniﬁcation may seem to require deduction. To unify (?x ?x) and ((a ?y c) (a b ?z)), for example, the algorithm must infer that ?x should be (a b c), ?y should be b, and ?z should be c. We may think of this process as solving a set of equations among the paern components. In general, these are simultaneous equations, which may require substantial manipulation to solve.71 For example, unifying (?x ?x) and ((a ?y c) (a b ?z)) may be thought of as specifying the simultaneous equations 71 In one-sided paern matching, all the equations that contain paern variables are explicit and already solved for the unknown (the paern variable). 622 ?x = (a ?y c) ?x = (a b ?z) ese equations imply that (a ?y c) = (a b ?z) which in turn implies that a = a, ?y = b, c = ?z, and hence that ?x = (a b c) In a successful paern match, all paern variables become bound, and the values to which they are bound contain only constants. is is also true of all the examples of uniﬁcation we have seen so far. In general, however, a successful uniﬁcation may not completely determine the variable values; some variables may remain unbound and others may be bound to values that contain variables. Consider the uniﬁcation of (?x a) and ((b ?y) ?z). We can deduce that ?x = (b ?y) and a = ?z, but we cannot further solve for ?x or ?y. e uniﬁcation doesn’t fail, since it is certainly possible to make the two paerns equal by assigning values to ?x and ?y. Since this match in no way restricts the values ?y can take on, no binding for ?y is put into the result frame. e match does, however, restrict the value of ?x. Whatever value ?y has, ?x must be (b ?y). A binding of ?x to the paern (b ?y) is thus put into the frame. If a value for ?y is later determined and added to the frame (by a paern match or uniﬁcation that is required to be consistent with this frame), the previously bound ?x will refer to this value.72 72 Another way to think of uniﬁcation is that it generates the most general paern 623 Applying rules Uniﬁcation is the key to the component of the query system that makes inferences from rules. To see how this is accomplished, consider pro- cessing a query that involves applying a rule, such as (lives-near ?x (Hacker Alyssa P)) To process this query, we ﬁrst use the ordinary paern-match procedure described above to see if there are any assertions in the data base that match this paern. (ere will not be any in this case, since our data base includes no direct assertions about who lives near whom.) e next step is to aempt to unify the query paern with the conclusion of each rule. We ﬁnd that the paern uniﬁes with the conclusion of the rule (rule (lives-near ?person-1 ?person-2) (and (address ?person-1 (?town . ?rest-1)) (address ?person-2 (?town . ?rest-2)) (not (same ?person-1 ?person-2)))) resulting in a frame specifying that ?person-2 is bound to (Hacker Alyssa P) and that ?x should be bound to (have the same value as) ?person-1. Now, relative to this frame, we evaluate the compound query given by the body of the rule. Successful matches will extend this frame by providing a binding for ?person-1, and consequently a value for ?x, which we can use to instantiate the original query paern. In general, the query evaluator uses the following method to apply a rule when trying to establish a query paern in a frame that speciﬁes bindings for some of the paern variables: that is a specialization of the two input paerns. at is, the uniﬁcation of (?x a) and ((b ?y) ?z) is ((b ?y) a), and the uniﬁcation of (?x a ?y) and (?y ?z a), discussed above, is (a a a). For our implementation, it is more convenient to think of the result of uniﬁcation as a frame rather than a paern. 624 • Unify the query with the conclusion of the rule to form, if suc- cessful, an extension of the original frame. • Relative to the extended frame, evaluate the query formed by the body of the rule. Notice how similar this is to the method for applying a procedure in the eval/apply evaluator for Lisp: • Bind the procedure’s parameters to its arguments to form a frame that extends the original procedure environment. • Relative to the extended environment, evaluate the expression formed by the body of the procedure. e similarity between the two evaluators should come as no surprise. Just as procedure deﬁnitions are the means of abstraction in Lisp, rule deﬁnitions are the means of abstraction in the query language. In each case, we unwind the abstraction by creating appropriate bindings and evaluating the rule or procedure body relative to these. Simple queries We saw earlier in this section how to evaluate simple queries in the absence of rules. Now that we have seen how to apply rules, we can describe how to evaluate simple queries by using both rules and asser- tions. Given the query paern and a stream of frames, we produce, for each frame in the input stream, two streams: • a stream of extended frames obtained by matching the paern against all assertions in the data base (using the paern matcher), and 625 • a stream of extended frames obtained by applying all possible rules (using the uniﬁer).73 Appending these two streams produces a stream that consists of all the ways that the given paern can be satisﬁed consistent with the original frame. ese streams (one for each frame in the input stream) are now all combined to form one large stream, which therefore consists of all the ways that any of the frames in the original input stream can be extended to produce a match with the given paern. The query evaluator and the driver loop Despite the complexity of the underlying matching operations, the sys- tem is organized much like an evaluator for any language. e proce- dure that coordinates the matching operations is called qeval, and it plays a role analogous to that of the eval procedure for Lisp. qeval takes as inputs a query and a stream of frames. Its output is a stream of frames, corresponding to successful matches to the query paern, that extend some frame in the input stream, as indicated in Figure 4.4. Like eval, qeval classiﬁes the diﬀerent types of expressions (queries) and dispatches to an appropriate procedure for each. ere is a procedure for each special form (and, or, not, and lisp-value) and one for simple queries. e driver loop, which is analogous to the driver-loop procedure for the other evaluators in this chapter, reads queries from the terminal. For each query, it calls qeval with the query and a stream that consists 73 Since uniﬁcation is a generalization of matching, we could simplify the system by using the uniﬁer to produce both streams. Treating the easy case with the simple matcher, however, illustrates how matching (as opposed to full-blown uniﬁcation) can be useful in its own right. 626 of a single empty frame. is will produce the stream of all possible matches (all possible extensions to the empty frame). For each frame in the resulting stream, it instantiates the original query using the values of the variables found in the frame. is stream of instantiated queries is then printed.74 e driver also checks for the special command assert!, which sig- nals that the input is not a query but rather an assertion or rule to be added to the data base. For instance, (assert! (job (Bitdiddle Ben) (computer wizard))) (assert! (rule (wheel ?person) (and (supervisor ?middle-manager ?person) (supervisor ?x ?middle-manager)))) 4.4.3 Is Logic Programming Mathematical Logic? e means of combination used in the query language may at ﬁrst seem identical to the operations and, or, and not of mathematical logic, and the application of query-language rules is in fact accomplished through a legitimate method of inference.75 is identiﬁcation of the query lan- guage with mathematical logic is not really valid, though, because the 74 e reason we use streams (rather than lists) of frames is that the recursive appli- cation of rules can generate inﬁnite numbers of values that satisfy a query. e delayed evaluation embodied in streams is crucial here: e system will print responses one by one as they are generated, regardless of whether there are a ﬁnite or inﬁnite number of responses. 75 at a particular method of inference is legitimate is not a trivial assertion. One must prove that if one starts with true premises, only true conclusions can be derived. e method of inference represented by rule applications is modus ponens, the familiar method of inference that says that if A is true and A implies B is true, then we may conclude that B is true. 627 query language provides a control structure that interprets the logical statements procedurally. We can oen take advantage of this control structure. For example, to ﬁnd all of the supervisors of programmers we could formulate a query in either of two logically equivalent forms: (and (job ?x (computer programmer)) (supervisor ?x ?y)) or (and (supervisor ?x ?y) (job ?x (computer programmer))) If a company has many more supervisors than programmers (the usual case), it is beer to use the ﬁrst form rather than the second because the data base must be scanned for each intermediate result (frame) produced by the ﬁrst clause of the and. e aim of logic programming is to provide the programmer with techniques for decomposing a computational problem into two separate problems: “what” is to be computed, and “how” this should be computed. is is accomplished by selecting a subset of the statements of mathe- matical logic that is powerful enough to be able to describe anything one might want to compute, yet weak enough to have a controllable procedural interpretation. e intention here is that, on the one hand, a program speciﬁed in a logic programming language should be an eﬀec- tive program that can be carried out by a computer. Control (“how” to compute) is eﬀected by using the order of evaluation of the language. We should be able to arrange the order of clauses and the order of sub- goals within each clause so that the computation is done in an order deemed to be eﬀective and eﬃcient. At the same time, we should be able to view the result of the computation (“what” to compute) as a simple consequence of the laws of logic. Our query language can be regarded as just such a procedurally in- terpretable subset of mathematical logic. An assertion represents a sim- 628 ple fact (an atomic proposition). A rule represents the implication that the rule conclusion holds for those cases where the rule body holds. A rule has a natural procedural interpretation: To establish the conclusion of the rule, establish the body of the rule. Rules, therefore, specify com- putations. However, because rules can also be regarded as statements of mathematical logic, we can justify any “inference” accomplished by a logic program by asserting that the same result could be obtained by working entirely within mathematical logic.76 Infinite loops A consequence of the procedural interpretation of logic programs is that it is possible to construct hopelessly ineﬃcient programs for solving certain problems. An extreme case of ineﬃciency occurs when the sys- tem falls into inﬁnite loops in making deductions. As a simple example, suppose we are seing up a data base of famous marriages, including (assert! (married Minnie Mickey)) If we now ask (married Mickey ?who) 76 We must qualify this statement by agreeing that, in speaking of the “inference” accomplished by a logic program, we assume that the computation terminates. Unfor- tunately, even this qualiﬁed statement is false for our implementation of the query lan- guage (and also false for programs in Prolog and most other current logic programming languages) because of our use of not and lisp-value. As we will describe below, the not implemented in the query language is not always consistent with the not of math- ematical logic, and lisp-value introduces additional complications. We could imple- ment a language consistent with mathematical logic by simply removing not and lisp- value from the language and agreeing to write programs using only simple queries, and, and or. However, this would greatly restrict the expressive power of the language. One of the major concerns of research in logic programming is to ﬁnd ways to achieve more consistency with mathematical logic without unduly sacriﬁcing expressive power. 629 we will get no response, because the system doesn’t know that if A is married to B, then B is married to A. So we assert the rule (assert! (rule (married ?x ?y) (married ?y ?x))) and again query (married Mickey ?who) Unfortunately, this will drive the system into an inﬁnite loop, as follows: • e system ﬁnds that the married rule is applicable; that is, the rule conclusion (married ?x ?y) successfully uniﬁes with the query paern (married Mickey ?who) to produce a frame in which ?x is bound to Mickey and ?y is bound to ?who. So the inter- preter proceeds to evaluate the rule body (married ?y ?x) in this frame—in eﬀect, to process the query (married ?who Mickey). • One answer appears directly as an assertion in the data base: (married Minnie Mickey). • e married rule is also applicable, so the interpreter again eval- uates the rule body, which this time is equivalent to (married Mickey ?who). e system is now in an inﬁnite loop. Indeed, whether the system will ﬁnd the simple answer (married Minnie Mickey) before it goes into the loop depends on implementation details concerning the order in which the system checks the items in the data base. is is a very sim- ple example of the kinds of loops that can occur. Collections of interre- lated rules can lead to loops that are much harder to anticipate, and the appearance of a loop can depend on the order of clauses in an and (see 630 Exercise 4.64) or on low-level details concerning the order in which the system processes queries.77 Problems with not Another quirk in the query system concerns not. Given the data base of Section 4.4.1, consider the following two queries: (and (supervisor ?x ?y) (not (job ?x (computer programmer)))) (and (not (job ?x (computer programmer))) (supervisor ?x ?y)) ese two queries do not produce the same result. e ﬁrst query begins by ﬁnding all entries in the data base that match (supervisor ?x ?y), and then ﬁlters the resulting frames by removing the ones in which the value of ?x satisﬁes (job ?x (computer programmer)). e second query begins by ﬁltering the incoming frames to remove those that can satisfy (job ?x (computer programmer)). Since the only incoming frame is empty, it checks the data base to see if there are any paerns that satisfy (job ?x (computer programmer)). Since there generally 77 is is not a problem of the logic but one of the procedural interpretation of the logic provided by our interpreter. We could write an interpreter that would not fall into a loop here. For example, we could enumerate all the proofs derivable from our assertions and our rules in a breadth-ﬁrst rather than a depth-ﬁrst order. However, such a system makes it more diﬃcult to take advantage of the order of deductions in our programs. One aempt to build sophisticated control into such a program is described in deKleer et al. 1977. Another technique, which does not lead to such serious control problems, is to put in special knowledge, such as detectors for particular kinds of loops (Exercise 4.67). However, there can be no general scheme for reliably preventing a system from going down inﬁnite paths in performing deductions. Imagine a diabolical rule of the form “To show P (x ) is true, show that P (f (x )) is true,” for some suitably chosen function f . 631 are entries of this form, the not clause ﬁlters out the empty frame and returns an empty stream of frames. Consequently, the entire compound query returns an empty stream. e trouble is that our implementation of not really is meant to serve as a ﬁlter on values for the variables. If a not clause is processed with a frame in which some of the variables remain unbound (as does ?x in the example above), the system will produce unexpected results. Similar problems occur with the use of lisp-value—the Lisp predicate can’t work if some of its arguments are unbound. See Exercise 4.77. ere is also a much more serious way in which the not of the query language diﬀers from the not of mathematical logic. In logic, we inter- pret the statement “not P ” to mean that P is not true. In the query sys- tem, however, “not P ” means that P is not deducible from the knowledge in the data base. For example, given the personnel data base of Section 4.4.1, the system would happily deduce all sorts of not statements, such as that Ben Bitdiddle is not a baseball fan, that it is not raining outside, and that 2 + 2 is not 4.78 In other words, the not of logic programming languages reﬂects the so-called closed world assumption that all relevant information has been included in the data base.79 Exercise 4.64: Louis Reasoner mistakenly deletes the outranked- by rule (Section 4.4.1) from the data base. When he real- izes this, he quickly reinstalls it. Unfortunately, he makes a slight change in the rule, and types it in as 78 Consider the query (not (baseball-fan (Bitdiddle Ben))). e system ﬁnds that (baseball-fan (Bitdiddle Ben)) is not in the data base, so the empty frame does not satisfy the paern and is not ﬁltered out of the initial stream of frames. e result of the query is thus the empty frame, which is used to instantiate the input query to produce (not (baseball-fan (Bitdiddle Ben))). 79 A discussion and justiﬁcation of this treatment of not can be found in the article by Clark (1978). 632 (rule (outranked-by ?staff-person ?boss) (or (supervisor ?staff-person ?boss) (and (outranked-by ?middle-manager ?boss) (supervisor ?staff-person ?middle-manager)))) Just aer Louis types this information into the system, De- Wi Aull comes by to ﬁnd out who outranks Ben Bitdiddle. He issues the query (outranked-by (Bitdiddle Ben) ?who) Aer answering, the system goes into an inﬁnite loop. Ex- plain why. Exercise 4.65: Cy D. Fect, looking forward to the day when he will rise in the organization, gives a query to ﬁnd all the wheels (using the wheel rule of Section 4.4.1): (wheel ?who) To his surprise, the system responds ;;; Query results: (wheel (Warbucks Oliver)) (wheel (Bitdiddle Ben)) (wheel (Warbucks Oliver)) (wheel (Warbucks Oliver)) (wheel (Warbucks Oliver)) Why is Oliver Warbucks listed four times? Exercise 4.66: Ben has been generalizing the query sys- tem to provide statistics about the company. For example, 633 to ﬁnd the total salaries of all the computer programmers one will be able to say (sum ?amount (and (job ?x (computer programmer)) (salary ?x ?amount))) In general, Ben’s new system allows expressions of the form (accumulation-function ⟨variable⟩ ⟨query pattern⟩) where accumulation-function can be things like sum, average, or maximum. Ben reasons that it should be a cinch to imple- ment this. He will simply feed the query paern to qeval. is will produce a stream of frames. He will then pass this stream through a mapping function that extracts the value of the designated variable from each frame in the stream and feed the resulting stream of values to the accumulation function. Just as Ben completes the implementation and is about to try it out, Cy walks by, still puzzling over the wheel query result in Exercise 4.65. When Cy shows Ben the sys- tem’s response, Ben groans, “Oh, no, my simple accumula- tion scheme won’t work!” What has Ben just realized? Outline a method he can use to salvage the situation. Exercise 4.67: Devise a way to install a loop detector in the query system so as to avoid the kinds of simple loops illus- trated in the text and in Exercise 4.64. e general idea is that the system should maintain some sort of history of its current chain of deductions and should not begin process- ing a query that it is already working on. Describe what kind of information (paerns and frames) is included in 634 this history, and how the check should be made. (Aer you study the details of the query-system implementation in Section 4.4.4, you may want to modify the system to in- clude your loop detector.) Exercise 4.68: Deﬁne rules to implement the reverse op- eration of Exercise 2.18, which returns a list containing the same elements as a given list in reverse order. (Hint: Use append-to-form.) Can your rules answer both (reverse (1 2 3) ?x) and (reverse ?x (1 2 3)) ? Exercise 4.69: Beginning with the data base and the rules you formulated in Exercise 4.63, devise a rule for adding “greats” to a grandson relationship. is should enable the system to deduce that Irad is the great-grandson of Adam, or that Jabal and Jubal are the great-great-great-great-great- grandsons of Adam. (Hint: Represent the fact about Irad, for example, as ((great grandson) Adam Irad). Write rules that determine if a list ends in the word grandson. Use this to express a rule that allows one to derive the relationship ((great . ?rel) ?x ?y), where ?rel is a list ending in grandson.) Check your rules on queries such as ((great grandson) ?g ?ggs) and (?relationship Adam Irad). 4.4.4 Implementing the ery System Section 4.4.2 described how the query system works. Now we ﬁll in the details by presenting a complete implementation of the system. 635 4.4.4.1 The Driver Loop and Instantiation e driver loop for the query system repeatedly reads input expressions. If the expression is a rule or assertion to be added to the data base, then the information is added. Otherwise the expression is assumed to be a query. e driver passes this query to the evaluator qeval together with an initial frame stream consisting of a single empty frame. e result of the evaluation is a stream of frames generated by satisfying the query with variable values found in the data base. ese frames are used to form a new stream consisting of copies of the original query in which the variables are instantiated with values supplied by the stream of frames, and this ﬁnal stream is printed at the terminal: (define input-prompt ";;; Query input:") (define output-prompt ";;; Query results:") (define (query-driver-loop) (prompt-for-input input-prompt) (let ((q (query-syntax-process (read)))) (cond ((assertion-to-be-added? q) (add-rule-or-assertion! (add-assertion-body q)) (newline) (display "Assertion added to data base.") (query-driver-loop)) (else (newline) (display output-prompt) (display-stream (stream-map (lambda (frame) (instantiate q frame 636 (lambda (v f)